OUASE- SFATEC DEFORMATION OF
A VRSGOELASTIC PMTE SUPPORTED ON A

Thesis for-the Degree off'Phgo. '
' mam STATE RNWERsrry
EUGENEL MARVIN ’
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thesis entitled

QUASI-STATIC DE FORMATION OF
A VISCOE LASTIC PLATE SUPPORTED ON A
POROUS ELASTIC FLUID-FILLED HALF-SPACE

presented by

Eugene L. Marvin

has been accepted towards fulfillment
of the requirements for

PhD nge in Engineering

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Major professor

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ABSTRACT

QUASI-STATIC DEFORMATION OF
A VISCOELASTIC PLATE SUPPORTED ON A
POROUS ELASTIC FLUID-FILLED HALF-SPACE

by Eugene L. Marvin

A method of analysis is presented for the investigation of developed
stresses, pore pressures, and displacements in a pavement structure con-
sisting of a flexible pavement supported on a water-saturated soil foundation.

A mathematical model of the pavement structure is constructed using
ideal materials. The flexible pavement surface is replaced by an imper—
meable, linear viscoelastic, thin plate of infinite extent in the model. A
porous elastic solid saturated with an incompressible, viscous fluid is used
to simulate the saturated soil foundation. A uniform circular load is placed
at the plate surface to approximate a vehicle wheel load. The mathematical
model is analyzed to obtain an approximation of the deformation that would
occur if the real pavement structure were subjected to a uniform circular
load.

Analysis of the model is reduced to the solution of a linear initial-
boundary value problem. The initial-boundary value problem is solved using
iterated Laplace-Hankel integral transformations. The transformed solution
images are inverted to obtain physically meaningful solutions to the problem

using numerical methods. The Hankel transform inversion is performed

Eugene L. Marvin

approximately by numerical integration yielding values of the Laplace trans-
form of the solution at discrete points. The Laplace transform data is then
used to construct a generalized Fourier series approximation of the time-
dependent solution.

A computer program was developed that utilizes the inversion algor-
ithm to invert the transformed solution functions. Stresses and displacement
at any geometric point in the flexible pavement structure may be obtained as
functions of time using the developed computer program.

Measured material properties were used to construct a numerical solu-
tion for the response to load of a hypothetical pavement structure. The results

of the analysis are presented.

QUASI-STATIC DEFORMATION OF
A VISCOELASTIC PLATE SUPPORTED ON A
POROUS ELASTIC FLUID-FILLED HALF-SPACE

BY

_/
9'

Eugene Ii. :Marvin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILCBOPHY
Department of Metallurgy, Mechanics, and Materials Science

June 1971

ACKNOWLEDGMENTS

The author wishes to express his appreciation to Dr. Robert W. Little,
Professor of Mechanics, who served as chairman of the authors graduate
studies committee and to Dr. George E. Mase, Professor of Mechanics,

Dr. Merle C. Potter, Associate Professor of Mechanical Engineering, Dr.
David H. Y. Yen, Associate Professor of Mechanics and Mathematics, and
Dr. Chi Y. Lo, Assistant Professor of Mathematics who served as members
of the committee.

The author wishes to thank Alan R. Friend of the Michigan Department
of Highways who worked with the author in developing the computer program

that is presented in Appendix A.

ii

ACKNOWLEDGMENTS 0.000.... ........ OOOOOOOOOOOOOOOO. O O
LISTOFFIGURES.. ........... .. ............... .....
LIST OF SYMBOLS (Roman) .............. . ........... . ............
LISTOFSYMBOIS(Greek)..... ..... ..................
CHAPTER
I. INTRODUCTION . ....... . . . . ............ . . ........ . . .
II. THEMATHEMATICALMODEL.................. ..
III. MECHANICS OF INTERACTING MEDIA . . . . . . ....... . ......
3.1 Fundamental Properties.................... .......
3. 2 Field Equations for Interacting Media . . . . . . . . . .
3. 3 Incompressibility Condition . . . . . . . . . . . ..... . ..... .
3. 4 Constitutive Equations. . . . . . . ...... . .................
IV. GOVERNING DIFFERENTIAL EQUATIONS OF PORO-
ELASTICITY
4. 1 The Differential Equations of Poro-Elasticity . . . . . . . . . .
4. 2 Axially Symmetric Deformation - Displacement
Generating Functions...................... ......
4. 3 Arbitrariness of E and 5 Functions . . .............
V. THE VISCOELASTIC MATERIAL .
VI. VISCOELASTIC THIN PLATE THEORY

6.1
6.2

TABLE OF CONTENTS

The Viscoelastic Plate Deflection Equation . .
Solution of Plate Equation. . . . . . . . . . . . . . . . . .

iii

11
12
16
18
22
28

28

31
33

35

40

4O
43

VII. THE PIJATE ON THE HALF-SPACE o o o o o o oooooo o o o o o ccccc 46
7.1 BoundaryConditions............... ................. 46
7.2 Initial Condition on v2: . .. .. .................. 49
7. 3 The Initial-Boundary Value Problem . . . .............. 52
VIII. SOLUTION OF INITIAL-BOUNDARY VALUE PROBLEM ..... 54

8. 1 Solution of Differential Equations Using Transform

MethOdSooooooooooooooooooooooooooooooooo ...... .0 54

8. 2 Determining Laplace-Hankel Transforms ........... . . 56

8. 3 Determining On . O" , 099 , and u, .............. 60

IX. INVERSION OF LaPLACE-HANKEL TRANSFORMS ......... 64

901 General Algorithm 000.....00.0000000000000000. ..... 64

9.2 Numerical Inversion of the Laplace Transform ........ 65

9.3 ImprovingConvergence............................. 67

9. 4 Selecting [m Points . .............. . . . ..... . ........ 68

X . EXAMPLE PROBLEM . . ................................. 73

XI 0 CONCLUSION 0000000000 0 0 .......... . . 0 0 0 0 000000 0 0 0 0 . 0 0 0 . 78

1101 summary.0.0.0000000000000.000. 000000000 .0........ 78

1102 Further Study.00.00 000000 00....00... 00000 0.. 0000000 79

REFERENCES 0000000 000.000 000000000 000.00...00.000.000.0000.000 81
APPENDIX

A. COMPUTER PROGRAM LISTING AND USER GUIDE . . ....... 83

B. DETERMINING MATERIAL PROPERTIES ........ . . . . . . . . . . 126

C. INITIAL AND STEADY STATE SOLUTIONS .......... . . . . . . . 135

iv

Figure

10

11

12

LIST OF FIGURES

Mathematical model .......
Typical stress relaxation function . . . . . . . . ....... . . . . . . . . . .
Multi step-function approximations of strain loading . . . . . . . . .

The viscoelastic plate ................ . . . ............... . .

Convergence comparison, Fourier partial sums, and Fejer

Sums00.....0...0000.......000........0....0.00..0.000..

Convergence comparison, Fourier partial sums, and

meanofsucceedingsums.. ...... ......

Numerical inversion results, test function . . ..... . . . . . . . . .

Example pavement structure......................... .....

Partial fluid stress at origin versus time ............ . ..... .

Partial fluid stress variation with depth at centerline of load,

time:zero.0........00.0.0.0.0..00..0.0. ....... .0000000

Surface deflection at origin versus time .............. . .....

Surface deflection at radial distance of 10 inches from origin

versus time...00....0.0....0...0.000...0.000.0...0....00

38

38

44

69

69

72

74

76

76

77

77

LIST OF SYMBOLS

Roman Characters

A - Total surface area.

AP " Pore area at surface.

0 , 0| , 02 , O3 " Physical constants of pom-elasticity.

b "' Load radius.

8| , 82 , B3 " Functions of Laplace and Hankel variables.
C "' Diffusion constant.

CI , C2 , C3 . C4 , C5 . C6 , C7 ' Functions of Laplace and Hankel variables.
Cm" " Matrix of coefficients used in constructing orthonormal functions.
dij ' Rate of deformation tensor for solid in mixture.

0 (i) " Plate stiffness function.

D " Plate geometric stiffness constant.

dl . 6" .dz , 6.2 .d3 . d'3 ' Nnctions of transform variables.

0” - Small strain tensor of solid.

3, , 39, 31 " Unit vectors.

E - Displacement generating function.

f - Porosity.

ff " Acceleration vector of solid particles.

fij " Rate of deformation tensor of fluid.

Fi " Body force per unit mass on solid.

9i ’ Acceleration vector of fluid particle.

Ci - Body force per unit mass on fluid particle.

h " Plate thickness.

vi

H (I) " Heaviside unit step function.

JV " Bessel function of first kind and V order.
k " Function of Hankel variable, I] .

K - Function of Hankel variable, n, .

{1m} " Sequence of real numbers.

(”In ,‘Z‘m " Solid and fluid mass elements in mixture.

MX’ My, Mxy -' Moment resultants per unit length along edge of plate.
"k - Unit vector normal to arbitrary surface at a point.

P - An arbitrary sealer function.

P '" The partial fluid pressure in the interacting mixture.

90“) " Sealer time dependent function.

q - Load pressure.

dq "' Heat input per unit mass of mixture.

q", q' - Stresses applied to positive and negative sides of plate paralled to Z axis.

0x . Q, - Shear resultants per unit length along edge of plate.

I' "' Radial co-ordinate.

R "' Pore compressibility constant.

R (I) - Viscoelastic material time operator.

Sm '- Total stress tensor for interacting media.

8| , .2 - Interacting continua.

8 " Laplace transform variable, or specific entropy per unit mass.
8 - Displacement generating function.

f - Time variable.

AI -' Finite time increment.

vii

T "" Absolute temperature.

0i " Velocity vector of solid particle in mixture.
[1' , “I - Physical components of displacement vector ina'and szdirections.
u - In-plane plate displacement inX direction.
V — Volume of interacting mixture.

Vp - Pore volume.

9‘ — Velocity vector of fluid particle in mixture.

V. " Volume of solid material contained in mixture.

V - Lateral displacement in plane of plate inydirection.

u - Plate deflection in 2 direction.

‘0 - Iterated Laplace-Hankel zero order transform of plate deflection.
Xi - Position Vector of solid particle in interacting mixture.

Xi - Initial co-ordinate position of solid particles prior to deformation.
’i - Position vector fluid particle in interacting mixture.

Y] " initial co-ordinate position of fluid particle prior to deformation
of interacting mixture.

2 "" Vertical co-ordinate.

Z - Function of Hankel variable, '1 .

viii

LIST OF SYMBOLS

Greek Characters

0|, 02, a4, 05’ 06, 08- Materials constants.
y - Fluid density function of interacting mixture.

5 - Unjacketed compressibility constant.
5'3 - Kronecker delta.

Eli - Strain tensor in viscoelastic plate.

8'“ - Deviatoric strain tensor in viscoelastic plate.

§ - Measured fluid outflow per unit volume.

fl - Hankel transform variable.

k - Jacketed compressibility constant.

AI,A2- Constants.

11- Variable.

V - Poisson's ratio of plate material, or order of Bessel Function.
17. - Diffusive force.

17". - Partial fluid stress tensor

‘ P - Initial density of fluid or solid prior to mixing.

- Density of fluid or solid reckoned per unit volume of mixture prior to loading.
‘ ’p - Density of fluid or solid reckoned per unit volume of mixture at any time, t.
on — Solid partial stress tensor. Also, viscoelastic material stress tensor.
Oik - Deviatoric stress tensor for viscoelastic material.

1: - A dummy variable of integration.

¢ - Stress relaxation function for viscoelastic material.

I

INTRODUCTION

The objective of this research has been to obtain the solution of the problem
of loading of a quasi-static plate on a deformable foundation. This problem arose
in the course of research being conducted to determine methods for structurally
analyzing highway pavement systems; a project sponsored by the Michigan De-
partment of State Highways in cooperation with the Federal Highway Adminis-
tration under the Highway Planning and Research Program. The plate on an
infinite half-space was proposed to mathematically model the response to load
of a flexible pavement supported on a water-saturated soil foundation.

The work utilizes a stress distribution theory which accounts for inter-
action of water and solid materials in the soil foundation under the pavement
and the viscous time-dependent effect in the pavement material. The viscous
effect is included in order to make the mathematical model of the pavement
structure as realistic as feasible as the viscous time effect is very significant
in bituminous concrete. However, the primary purpose of the research was
to develop an analytical tool for investigating water-solid interaction in sat-
urated pavement foundations subjected to external loads. A uniform circular
loading pattern was adopted because it approximates the load pattern of a

single wheel load and can be simulated experimentally by plate loading tests.

2

The main product of this research is the solution of the viscoelastic plate
on poro-elastic fluid filled half-space problem. In conducting the research nec-
essary to solve the defined problem two secondary new results were obtained.
The first of these was the determination of the initial condition on the poro-elastic
foundation that allows for compressibility in the elastic material making up the
skeletal porous structure of the medium. The second consisted of the develop-
ment of a numerical technique for inverting iterated Laplace-Hankel transform
functions.

Before proceeding with the mathematical analysis, it might be appropriate
to discuss the problem on purely physical grounds in order to illustrate the sig-
nificance of the study and its relationship to the overall problem of analyzing
flexible pavement structures. The net effect of the solid-fluid interaction in the
soil foundation is that the hydrostatic stress applied to the soil particles is less
than that applied to an element of the soil mass. The reason for this is that part
of the load applied to the soil is carried by the fluid. At points in the foundation
where the pressure on the fluid is large, the soil structure is weakened because
the ultimate undrained shear strength of the soil material decreases as the ef-
fective pressure on the solid particles decreases.

The solid-fluid interaction phenomenon is time-dependent and becomes
significant after some period of time. The nature of the decay of this tran-
sient effect is of interest because, as the interaction reaches a steady state
condition, the effective solid particle stresses approach the applied stresses
and the fluid pressure approaches zero. The reason the interaction dissipates

is that the fluid in the soil foundation tends to move from areas of high total

3
pressure to areas of low pressure. As the fluid moves out of a highly stressed
region, the solid particles in the element are required to carry a greater por-
tion of the load in order to maintain equilibrium.

The deformation of the foundation is time-dependent because, as the fluid
flows away from highly stressed regions, the soil decreases in volume and
settling occurs. If a stationary load is placed on the pavement structure for
a period of time, the pavement will continue to deform for some time after the
load is applied due to the solid-water interaction occurring in the foundation.
The viscous properties of the bituminous concrete cause this effect to be
magnified because, although the this material exhibits a relatively high initial
stiffness, it creeps under constant loading.

In actual service, loading that is applied to highway pavements is transient
in nature. The loading applied at a point in the foundation increases as a vehicle
approaches and decreases as it departs and this process is repeated many times
during the service life of the pavement. The net effect of the service loading
can be approximated by utilizing superposition techniques. In this research,
the effect of a uniform circular load--applied instantly and then held constant
indefinitely--is studied. By adding up the effect of several such loads applied
at different times, any smooth time-dependent loading can be approximated.
Therefore, it is apparent that the solution determined in this study is a build-
ing block which can be used to approximate more complicated time-depend-
ent pavement loadings if necessary.

The viscous property of the bituminous concrete is temperature sensitive.

However, isothermal deformation is assumed in the subsequent analysis

4
presented in this report and it is necessary to regard the deformation as occur-
ring at some specified temperature. Considering the service loading that is to
be ultimately approximated, it seems reasonable to assume that the temperature
of the pavement will remain constant during the passage of one service load. The
temperature distribution in the pavement slab can be expected to vary with depth;
however, the bituminous layer thickness is relatively thin (less than five inches)
in the pavement structures being considered in this study and, therefore, the
temperature gradient can be expected to be small.

This report is organized into eleven chapters. In Chapter II, a mathematical
model of the pavement structure is introduced. The ideal poro-elastic mate-
rial, which simulates the foundation in the model, is then discussed in Chapters
III and IV. To begin with, in Chapter III the mechanics of deformation for the
interacting media foundation are defined. ' In that chapter, the necessary field
equations and constitutive equations are presented. The system of defining
equations is reduced in Chapter IV by substituting the constitutive equations
into the equilibrium equations. Further reduction is accomplished by consid—
ering the axial symmetry of the foundation loading. Two unknown displacement
generating functions are introduced which define the axisymmetric deformation.
In Chapter V, the ideal linear viscoelastic material is discussed. This mate-
rial is used to simulate the bituminous pavement layer in the model. Constitu-
tive equations are presented that define the deformation response to load for
the viscoelastic material. Thin viscoealstic plate theory is introduced in Chap-
ter VI. A single differential equation defining the deformation response of the

plate is obtained by utilizing the continuum equilibrium equations, the

5
constitutive equations, and the geometry of the plate. By applying the methods of
operational calculus the plate equation is reduced to an algebraic expression
relating the iterated Laplace-Hankel transforms of the unknown plate deflection
and the foundation reaction.

In Chapter VII, an initial-boundary value problem is defined for the founda-
tion. The unknown foundation reaction is one boundary condition of the problem.
In Chapter VIII, the initial-boundary value problem is solved to obtain an ex-
pression for the Laplace-Hankel transform of the foundation reaction. The ex-
pression that is obtained is combined with that determined in Chapter VI to yield
explicit expressions for the transformed plate deflection and foundation reaction.
The transformed solutions of other unknown stresses and displacements in the
foundation are then determined using the deflection and foundation reaction
expressions.

In order to determine physically meaningful solutions, it is necessary to
invert the transformed solution images. This mater is dealt with in Chap-
ter IX. The inverse Hankel transformation is approximated using a numerical
integration algorithm. The inverse Laplace transformation is accomplished
by approximating the time-dependent solution with a generalized Fourier
series. Knowledge of the Laplace transform of the solution permits con-
struction of the series approximation.

In Chapter X, the results of the analysis are applied to a hypothetical pave-
ment structure using measured material property data. Numerical results are
given that include plate deflections, foundation pressures, and fluid pressure

distribution and decay curves. Finally, in Chapter XI, some conclusions and

6
recommendations are presented concerning the application and extension of the
results obtained in this study. The computer program developed to perform the
pavement analysis is included in the Appendix. The program is designed so that
other investigators may use it to perform stress analysis of flexible pavement
structures. Laboratory tests for determining the material constants required

are also discussed in the Appendix.

II

THE MATHEMATICA L MODE L

’A mathematical model of the flexible pavement structure may be con—
structed using ideal materials to simulate the actual physical materials. Once
this is accomplished, the model is then assumed to be subjected to loading, and
analyzed to obtain an approximation of the deformation that would occur if the
real structure were subjected to the same loading.

The mathematical model used to simulate the flexible pavement
structure in this study consists of a linear viscoelastic plate supported on a
linear poro-elastic solid saturated with an incompressible fluid as shown in
Figure 1. The plate is of infinite extent and the foundation occupies the half-
space lying directly below the plate. A vertical step load is applied to the plate
at zero time. The load is uniformly distributed over a circular area of radius b
on the plate surface. The model is considered to have been at rest and unstressed
prior to application of the load.

The following assumptions have been made concerning the physical
response of the model:

1) body forces are neglected (A body force solution may be super-
imposed on the results obtained. )

2) the deformation occurs at a constant temperature

3) the deformation of the model is quasi-static (independent of inertial
effects)

.TetoE EofianfiaE A 0.2%;

 

 

0.40m
U.Pm<JmIO¢Om

 

 

 

 

whit—n. 2.1... v A 034
th(n_uoum_> m<JDUEU :10“:an

 

 

9
4) the model is assumed to be at rest prior to application of the load

5) the displacements and velocities at all points in the model are small
so that linear theory is applicable

6) the viscoelastic plate material is assumed to be incompressible
7) the boundary between the plate and the foundation is frictionless and,
therefore, only normal stresses are transferred from the plate to the

solid

8) the vertical displacement of the plate is equal to that of the foundation
at the interface

9) the thin viscoelastic plate forms an impermeable boundary to fluid flow

10) the foundation consists of a linear, isotropic, homogeneous perfectly
elastic solid and an incompressible fluid

11) the foundation is homogeneous and has isotropic mechanical properties
(This condition is implied by assumption 10 but is stated separately to
emphasize that solid-fluid mixture is assumed to exhibit these properties
also.)

12) creeping flow of the fluid in the foundation is assumed. (Viscous ef-

fects on the flow through the pores are accounted for by introducing a per-

meabflity parameter.)

13) the shear stresses acting on the fluid are assumed to be small in com-

parison to the normal stresses and, therefore, may be neglected in the sol-

ution of stress distribution problems in the foundation.

Finally, it is observed that an alternative model consisting of an infinite
strip supported on a half-space foundation could be utilized rather then the infinite
plate on half-space model. Such a model would be more difficult to analyze be-
cause the conditions of support at the edges of the strip would need be considered.

Such sophistication does not seem to be warranted in modeling bituminous
concrete pavement structures, however, because for the most part the traffic

loads are confined to wheel tracks that are located several feet from the edge

of the pavement. Also, bituminous concrete pavements are flexible and probably

10
do not distribute significant amounts of the applied loads very for laterally. It
is therefore quite likely that the foundation reaction under the pavement becomes
insignificant within a lateral distance of a few load radii from the center of ap-
plication of the load, and at some distance from the edge of the pavement. For
these reasons it is thought that using an infinite plate model rather than an infinite

strip model provides a satisfactory approximation to the physical problem.

III

MECHANICS OF INTERACTING MEDIA

Biot [1] formulated a theory of deforming fluid filled porous media in
1939 to mathematically describe Terzahgi's porous solid model. Terzahgi [2 ]
proposed a porous fluid filled solid model to mechanically depict soil consolidation
under load. Previous models treated the earth as elastic and neglected moisture
and porosity effects. Terzahgi's one dimensional analysis of the model resulted
in a theory of consolidation which is used to predict the consolidation of soft clay
layers of soil underlying building foundations and embankments. The Biot theory
has been developed and applied by many researchers since its conception and has
become the classical theory of pom-elasticity. Recently, Paria [3 ] has dis-
cussed the Biot theory and its history of development.

The mechanics of interacting media treated here was formalized more
recently (1965) by Green and Naghdi [4] and is based on thermodynamic con-
siderations. This theory of continuum mechanics is quite general and ap-
Plicable to mixtures of gases, solids, and fluids. In the case of a mixture of
a Perfectly elastic solid and an incompressible fluid, the modern interacting media
theory reduces to the classical formulation of poro-elasticity. In 1968, Tabaddor [5]
discussed the development of the modern theory and its relationship with poro-

eIaBtICIty and theories of flow through rigid media. In the present paper,

the interacting media theory will be discussed for the case when the mixture
11

12

consists of a poro-elastic material and an incompressible fluid.

3. 1 Fundamental Properties

 

In the development of the mechanics of interacting media, certain basic
assumptions have been made [4, 5] and will be referred to here as basic
postulates.

Postulate I: At every point in a poro-elastic continuum there exists
both fluid and solid, neither of which may be isolated physically
from the mixture.

Physically, Postulate I can be examined by a microscopic phys-
ical model which has been proposed to describe poro-elastic material [1, 3] .
Microscopically, the material is assumed to consist of an elastic skeleton
containing many interconnected void spaces, or pores, which are filled with
fluid. A small material element is thus seen to always contain both
fluid and solid. However, when macroscopic structures of poro-elastic mate-
rials are viewed, the small element appears to be a point in the structure.
(For example, in the problem being considered here the macroscopic structure
is a half-space of infinite extent.) Therefore, it is assumed that the small
element of material can be considered a point in the macroscopic structure,
and in this sense there exists both fluid and solid at every point in the mix-
ture.

The mixture of solid and fluid at an arbitrary point in the mixture will

take on new density properties. The density properties of the mixture at a
point in the macroscopic body can be examined by first considering a small
finite element and then shrinking it to a point. The fluid in the element has
a true density of tab which is equal to its mass divided by its volume.

__.-__

An artificial fluid density mp can be defined by dividing the fluid mass

iii

in an unstressed poro-elastic element by the total volume of the element. This
density “’5 can be thought of as the apparent density of the fluid in the element
prior to application of load. Further, if the element is deformed such that its
volume changes, and possibly some of the fluid flows out of it, a new artifi-
cial density (zip may be defined as the ratio of the current fluid mass in the
element and the current volume of the element. (Zip can be cmsidered to
be apparent fluid density of the deformed poro-elastic material at time 1
Similarly, the elastic solid in the skeletal structure of the volume element can
be seen to have a real density “’3 , an artificial density “’5 when unstressed
and a density ”’p at firm i equal to its mass divided by the deformed vol-
ume of the element. Now, if the volume element is shrunk to a point, and it
is assumed both fluid and solid exist at the point, it must be assumed that the
densities defined above do not change.

In order to proceed with the development of the mechanics of interacting
media, it is necessary to state Postulate I mathematically. Considering a
mixture of two continua s' and 32 , that are in relative motion, the posi-

tion of each point in a. and in 92 are given by Eq. (1):
xi ' nix: .X2' x3,” . Yi ' 7|in .Y2.Y3. t). 5' I.2.3 (1)
where x. denotes the position of each point in s' and y. that of 32 .

Both position vectors xi and y; are expressed in Cartesian co—ordinates
and are measured from a common Cartesian reference frame. The co-
ordinates Xi and Yi refer to the original positions of the points in a. and
:2 prior to deformation. POstulate I states that every point in the mixture is
occupied simultaneously by a point in a. and one in s2 . Focusing on an

arbitrary point in the mixture it is apparent that some point that was

l‘l

originally at X. in 3' , and one that was at VI in 32 , now occupy the

arbitrary point in the mixture as indicated by Eq. (2):
‘j"|.i'l.2,3 (2)
Since Eq. (2) is true for any arbitrary point in the mixture, it is true through-

out the mixture at any given time i . It can be seen, therefore, that Eq. (2)

is mathematically equivalent to Postulate I.

Postulate II: The total stresses that are applied to an infinitesimal
volume element at a point in the poro-elastic body are equal to the
sum of the partial stresses on the solid and fluid continua at that
point.

Physically, this postulate can be understood by considering a small finite
element of the poro-elastic mixture which is subjected to surface traction stres-
ses on all sides. If the total force applied on any given face is divided into that
applied to the solid and that applied to the fluid, then two partial stress com-
ponents can be defined by dividing these two forces by the gross area of the
element surface. letting 5 “denote the total stress traction components ap-
plied to the cubic element of the poro-elastic mixture, it can be seen that par-
tial solid and fluid stress components 0'" and mkare related to the total stress

components by Eq. (3).
0'“, "’ ”it: ' 3n; (3)
Now if the cube is shrunk to a point, or more accurately, if it is assumed to

be infinitesimal relative to the poro—elastic body, then Oil: and 17", may be
thought of as the stresses applied to the two artificial continua that exist at
every point in the mixture according to Postulate I. The stresses on the two
continua, therefore, must satisfy Eq. (3) which is equivalent to the condition

stated in Postulate II.

15
Another basic property that is exhibited by the interacting media consid-

ered in this research is that of porosity. The poro-elastic mixture is assumed
to have a strong skeletal structure as opposed to a colloidal mixture of solid
and fluid having no pore structure. It is therefore possible to define the por-

osity f of an element of the skeletal structure as shown in Eq. (4):

, . Eve (4)
where V denotes the volume of the interconnected pores, or void spaces, and

p
v is the total volume of the element. An alternative definition of porosity

may be derived by considering the mean pore area of the surfaces that are
oriented parallel to one face of a cubic element of the skeletal structure. The

mean pore area AP is defined by Eq. (5):
h

Ap=%-/mtndz (5)

where h denotes the length of the element and m the ratio of pore area to total
area on the surfaces parallel to the z faces of the element. The volume vp can

be computed in terms of Ap as shown in Eq. (6).
Vp = Aph (6)
Substituting Eq. (6) into Eq. (4) yields an alternative expression of porosity in

terms of AP and the gross area of the element face A .

.52
f A (7)

This second definition of porosity is not used in the subsequent develop-
ment of the mechanics of poro-elasticity. It is used, however, in interpreting

laboratory tests which are performed to determine the material constants.

ii;
3.2 Field Equations for Interacting Media

 

The approach used by Green and Naghdi [4 j to establish field equations
assumed that the interacting mixture satisfied Postulates I and II, given in
3.1. They considered the thermodynamics of the interacting mixture and
adopted a third postulate.

Postulate III: The first law of thermodynamics as stated in Eq. (8)
applies to interacting media.

Rate of flow of

Rate of Increase Power of the ex- .
mechanical and non— 8
of energy in a = ternal forces on - ( )
mechanical energy
mass system. the mass system.

into the mass system.

An energy balance equation which expressed Eq. (8) in mathematical terms was
derived for a fixed arbitrary volume in space, and invariance conditions which
result when the effect of rigid body motions of the mixture are considered were
systematically applied. The equations of mass conservation and motion
derived using this technique are analogous to the well known field equa-
tions for single constituent continua, such as for example, the elastic solid.

The conservation of mass equation is given in Eq. (9):

 

pr m - Dub (2: ° - 9
5+ puk,k+ D? + pvhk '0 ( )

where 7%- is the material time derivative operator which measures

the rate of change of the property of any given particle of mate-

rial as observed from a fixed point in space. 0“,“ and Flak are

the divergences of the solid and fluid particle velocities at a spatial point
in the solid and fluid mixture. When no chemical action takes place between

the fluid and solid components in the mixture, each mass element is conserved and

l7

stronger continuity conditions hold.
Dflip

 

m - +m a 3 (10)
[“3 0' p k,k 0

Dub (2 ' (11

(2)m = F'— + )pvk.kgo )

The resulting equations of motion are shown in Eq. (12).

The new variables F;,G;.fi and 95 , introduced in Eq. (12), are, re-
spectively, the body forces per unit mass on the solid and fluid and the accel-
eration vectors of the solid and fluid. The last two terms in Eq. (12) vanish
according to the strong continuity conditions Eqs. (10) and (12). For quasi-
static deformations, the two acceleration terms on the right side of

Eq. (12) may also be dropped. In the absence of body forces and with quasi-
static deformation Eq. (12) reduced to Eq. (13) which is the form used in

this study.
(Uki ”’kibk =0 (13)

A diffusive force vector 1r: was introduced to combine certain force
terms that appear in the energy balance equation and are attributed to
interaction occurring between the two deforming media. 1ri is defined as

shown in Eq. (14):
ms-‘z-(O'M-fiki),k + '?"’p(Fi-fi)- 'é'Q’PiGi'Qi) (14)
In the absence of body forces and for quasi-static deformation, Eq. (14) reduces
to Eq. (15):
7’: 'liwki'fikibk (15)
Applying a rigid rotation invariance argument to the energy bal-

ance equation, Green and Naghdi showed that the total stress tensor must

 

(1) The commas appearing in Eq. (9) and (12) indicate partial differentiation
with respect to "It‘ Repeated indices indicate summation. These con-
ventions apply throughout the text.

18

be symmetric as shown in Eq. (16).
OH +7“ ”Gwyn“! (15)

The principle results of this summary that will be used in the subsequent
poro—elastic analysis are expressions (13) and (16) and definition (15) . These
are the quasi-static equilibrium equations, the total stress tensor symmetry
relation, and the diffusive force definition. To complete the formulation of
the mechanics it is necessary to obtain constitutive relationships for the
stresses and diffusive forces that will relate these variables to the deforma-
tion of the media. Before considering the formulation of constitutive equa-
tions, however, it is useful to consider what constraints are placed on the
deformation of the interacting mixture by the incompressibility of the pore
fluid.

3. 3 Incompressibility Condition

 

The fluid in the poro-elastic material is assumed to be incompressible.
However, this does not imply that the artificial fluid continuum with density
"’p is incompressible. Physically, this becomes apparent when a small finite
volume element of the poro-elastic material is considered. In such an e1-
ement, the fluid is contained in pores which are surrounded by an elastic
skeletal structure. Suppose, for example, that the cube faces are sealed so
that none of the fluid can escape. Then when the cube is compressed the elas-
tic skeleton will decrease in volume and the fluid fflled pores will become more
closely spaced. Therefore, the volume of the element can be decreased with-
out changing the actual fluid content in it, and the artificial density ”’1; which
is equal to the fluid content divided by the current volume of the cube can be

changed without removing any of the fluid from the element.

19

Consider as another example a permeable cube, and assume that
elastic material in the cube to be incompressible also. In this case, if the
cube is compressed for a period of time the skeletal structure will change
shape but its volume will remain the same. As the pressure is applied the
fluid will flow out of the pores, allowing the pores to decrease in volume.
Therefore, the artificial density "’p will change because the original elas-
tic material is now contained in an element of decreased volume. Thus, it
can be seen that incompressibility of the elastic skeletal material does not
imply incompressibility of the artificial elastic continuum. Other density
phenomena can be predicted for the artificial interacting continua; however,
further examples will not be pursued here as the two examples given serve
to motivate the analysis that follows. The derivation that is given here fol-
lows the work of Tabaddor [5] .

Consider a small unstressed element of poro-elastic material with
initial porosity f initial solid volume VI and volume V. at time t . The
volume V. is defined as the volume formed by the extreme boundary sur-
faces of the mixture. Although the actual volume of the elastic material in
(the skeletal structure is less than that of the cube, the artificial elastic solid
used for the continuum approximation has exactly the same volume as the

element of poro-elastic mixture. Since the elastic strains are assumed to

be small, the change in volume per unit cube of the artificial elastic mate-
rial may be taken equal to 'mm , the sum of the normal strains. It is, there-
fore, possible to express v. in terms of (7‘ as shown in Eq. (17):

.. 17
v| a \I| ( I 4- .mm) where 'mm 2 “m,m ( )

20

It is reasonable to assume further that there exists a linear relationship
between the element compressibility and volume change of the pores, after the
initial dilatation of the skeletal material has taken place. If such a relation-
ship is assumed and v is defined as the current pore volume per unit vol-

ume, then the pore volume may be expressed as shown in Eq. (18):
v = VI (3 (.mm - emmiol) 4- f) (18)

where emmh) denotes the initial value of ”mm at time zero when the load-
ing is applied, and R denotes the ratio of pore compressibility to total el-
ement compressibility. R depends on the geometry of the skeletal structure
and the nature of the material filling the pores. As the material is compressed,
the pores will become smaller, giving an indication of pore compressibility. How-
ever, the elastic skeletal structure will be compressed also and, therefore, the
total change in volume of the cubic sample will be greater than the change in pore
volume. If the pores are filled with an incompressible fluid, the pore volume
change will depend on the fluid flow from the element. In this case, strong
interaction occurs between the fluid and solid and the change in volume of
the skeletal structure is also dependent upon the flow of fluid from the el-
ement.

If the skeletal structure material is incompressible, the pore compres—
sibility equals the total compressibility and R 3| . Biot and others [1, fi]
have treated the analysis of such materials. In the present research, mate—
rial having a compressible skeletal structure, such that R will be less than
one and greater than zero. is analyzed.

It is desirable for analytical purposes to consider R as constant.

2.1

l’nsiulzltc IV states the assumed condition:
Postulate IV: The ratio of pore compressibility to total compres-
sibility of a small finite element of the porous, fluid filled solid is
assumed to be constant for times greater than zero.

From Postulate IV it is seen that the validity of the analysis will vary
between different porous media according to how nearly R approaches a con-
stant value for t>o and for how long it remains so. Therefore, in investiga-
ting specific materials, this limitation should be measured by experirnentally
observing the behavior of R .

Treating R as constant, we proceed to determine the constraint implied

by the pore fluid incompressibility. The total mass of fluid in the pores per

unit volume at time t can be expressed as shown in Eq. (19):

(25 v a (Zip v.
(”D V (19)
(2)5 V.

The second equality follows from the given expression. Similarly, the
initial apparent density of the artificial fluid can be expressed by the first
Eq. (20) which implies the second equality:

(2:5 .(afif

- 20
(2)5 3 :32 ( )
f

The ratio of current pore volume to current element volume can be expressed

in terms of fluid densities, or in terms of apparent solid strain as shown in

Eq. (21):?

(2)
{7,258 ' FT 9‘ (0mm ' Omani‘mm’” 4’ f ' “MW (21)

 

The final approximate equality was obtained from Eqs. (17) and (18) by
performing synthetic division, and neglecting the second order terms
in the result.

22
Equation (22) follows by equating the left and the right sides of Eq. (21), and

rearranging terms.
(2

fig. .mm(a-n+ :-R(omm(o» (22)

It is assumed that the deformation is isothermal, and that no chemical
reaction occurs in the mixture. The last assumption implies that the mass
elements of each component of the mixture are conserved during deformation.
According to these assumptions the current apparent fluid density mp can be
linearly related to the apparent density at time i 80 by a small sealer

function )’ as shown in Eq. (23):
(2b a (2b(°) + y (23)

Substituting Eq. (23) into the conservation of fluid mass equation (Eq. (11) of

3. 2) and neglecting the second order terms, results in Eq. (24):

a __( a!“ 3 __ “’5 3"): k
331 2b“) at (l+emm(o)) TL- (24)

If the partial time derivative is taken in Eq. (22), Eq. (25) results.

 

f 21 69mm
(‘53 at '(R'” a: (25)
Substituting Eq. (24) in Eq. (25) results in the desired constraint condition
Eq. (26). a
"It It (R-f) 30mm
—.I_ c .— ———.
a, f at (I +emm(0)) (26)
or3
Wink _,.._,_ (f-R) “mm s _o c”9mm
at f a: | a:
This equation states the condition that is forced on iakaLk and 22$ by
3

assuming that the pore fluid is incompressible.

3. 4 Constitutive Equations

 

Field equations that insure equilibrium and continuity of mass through-

out the interacting mixture have been given in 3.2. It is also necessary,

 

a The approximate equality on the left of the second Eq. (26) follows

from the assumption that the first invariant of the solid strains is
small, that is, much less than unity.

23

however, to relate the partial stress tensors at any arbitrary point in the mix-
ture to the deformation at that point. When this has been accomplished it will
be possible to relate deformations occurring in the interacting media to
stresses applied at the boundary of the media in terms of a boundary value
problem.

Tabaddor [5] has derived constitutive equations for a poro-elastic
mixture consisting of a perfectly elastic solid and an incompressible viscous
fluid for the case of small strain and strain rates. His work follows the more
general work of Green, Naghdi, and Stee1[4_, g]. The following fifth postulate
led to the results obtained:

Postulate V. The mixture of interacting media satisfies the second
law of thermodynamics.

The second law of thermodynamics postulates the existence of an entropy state

function swhich satisfies Eq. (27).

state 2
. 29.
As >/( T ) (27)
stars I
where As = change in specific entropy per unit mass
T = absolute temperature
dq = heat input per unit mass which is not an exact differential

The equality sign in Eq. (7) holds for reversible processes, and the inequality
sign for irreversible processes. The change in entropy is greater than that
produced by heat input in the irreversible case because internal entropy pro-
duction occurs due to dissipative processes, such as internal friction.

Green and Naghdi [4] constructed an entropy production inequality for a system

of interacting media undergoing a reversible or irreversible process. This was

24

done by examining entropy production of the mass system instantly occupying
an arbitrary volume V in space. Since the expression obtained applies for
any arbitrary choice of volume, the integral form of inequality may be local-
ized to an arbitrary point. Green and Steel [6] mathematically expressed the
condition at a point in terms of a localized entropy production inequality. In
order to make use of the inequality, it is necessary to construct the general
form of the constitutive equations.

Green and Steel, in postualting constitutive equations for a mixture con—
sisting of a non—linear elastic solid and viscous Newtonean fluid, relied
on the known constitutive properties of each component for inspiration [6]. .
The condition of isotropy was imposed on the assumed equations. The
assumed constitutive relationships were then substituted into the entropy
production inequality. This resulted in an inequality which contained sev-
eral unknown state variables having undetermined coefficient functions. By
arbitrarily varying the state variables one at a time, constraints on the
unknown coefficients were obtained. These conditions were then substituted
back into the assumed constitutive relationships. In this manner, Green and
Steel succeeded in obtaining constitutive equations for the non-linear elastic
and viscous fluid mixture. They, then linearized the theory for the case of
small elastic strains.

Tabaddor [5'] investigated the same mixture with the one additional phys-
ical restriction that the fluid be incompressible. He derived a constraint
on the rate of deformation tensors as a consequence of the fluid incom-
pressibility. The incompressibility condition was presented in 3. 3 . Tabaddor

substituted his incompressibility Eq. (28) in the entropy production inequality.

25
av", an,“ E
—a_"m + 0' 3' ' O (28)
He then applied the procedure discussed above to determine the constraints
placed on the unknown coefficients of the state variables in the entropy in-
equality. This resulted inaset of constitutive equations for a non-linear elastic
solid and a viscous incompressible fluid. Tabaddor simplified his constitu-
tive equations further by introducing the infinitestimal strain assumption.
The resulting constitutive equations are:
“In
“ik ' oki'al 5n: ”04'“: ‘5’ ”mm 5m + 2 ‘°I* °2ieik
a -
+(a8 *7}, 7 Mt + “up 5n:
- - a
”ik' ”ki"(2508’3*[‘2’0 “6*(‘2’Wm32i7 (29)
(I)-
+‘2’ptaz- fazlemm) but * M in MR t 2"2 fit:
l

60'
”i' “31"

t

0: L20

where at!!! av“ i
'ik ' 2‘ at + a: ’
and 0n: ! 21“,” + 0m)

If it is assumed that there are no initial stresses on the continua prior to ap-

plication of the load at time zero, then Eq. (29) reduces to the following:

“in ' Uni ' ‘94'mm + “BY“NR + 202°“:

1rik . .(25 as)! +‘25 (180mm ) a", + All" in 4”“2"“: (30)
cmi an

In order to reduce the constitutive equations of the fluid, it is assumed that
the viscous terms, A , f" and 2A2 f", are small in comparison to the

hydrostatic pressure. Neglecting these terms in 17", yields Eq. (31):

26

P ' (“’5 as 7 + ‘25 as 'mml (31)
where 1r“, has been replaced by '96", because the normal components
of 1r", are equal and the shear components vanish.
A change of variable can be accomplished by considering Eq. (24) of 3. 3
which reduces to Eq. (32):
gel ' "2’5 if? (32)

because ‘mm is assumed to be small.

Integrating this expression and noting that 7 n 0 at t a 0 yields Eq. (33).
7 = “PM. (33)
Substituting Eq. (33) for y in Eqs. (30) and (31) yields Eq. (34).

Gilt =(“49:11:11 ' “8‘25 Vm,m) 8it: + 202 9ii:

. (34)
p ’ "(GGWFF Vm,m ‘aB‘ZF’ 9mm) 6ik

The constitutive equations (34) are same as those given by Biot and Willis in

equation(1) of reference [II -

The variable Vm,m can be eliminated from the first constitutive
equation (34) by solving for "min in the second equation and then treating
p as unknown.

0", ' (Org—9612) °mm 55). +6.61%) pin. + 2 a2 en. (35)

It is shown in Appendix B, that by experimental determination of the
physical constants of poro-elasticity, Eq. (36) is valid for the problem being

studied here.

a
0| "' ' 6335 (36)

Using Eq. (35) and defining 03 as the coefficient of 9mm in Eq. (35) results

in the form of constitutive equations that are used in this study.

27

on: = '°l95ik+ 2“'2 Oak + 03 0mm m
”it: "Nil:

60' GT
773 " 0(31‘ - Til.)

Equations (37) relate the deformation occurring at any arbi-

(37)

trary point in the interacting media foundation to the state of stress there at

times greater than zero. Since y is zero initially at t: o , the constitutive

equations for 0'”, and p reduce to the following at the instant of loading.
0:). ' “flaw in. + 202°“:

(38)
P 3 “8‘25 °mm 8ik

IV

GOVERNING DIFFERENTIAL EQUATIONS OF PORO—ELASTICITY

4. 1 The Differential Equations of Pom-Elasticity

 

Equations (1) through (5) summarize the results obtained in Chapter III:

(Uki+7’ki)'k ' 0 a)”
oki'oik"°lp5ik+2°2°ik+°3°mm5ik (2)
"ii ' ”ik' ‘9‘": (3)
1r, -o(%‘-’;'l la?) (4)
in "W239.” ‘5’

These equations govern the quasi-static deformation of an interacting continua
mixture consisting of a linear elastic solid and an incompressible fluid at times
greater than zero. Equations (1) are the equilibrium equations of the continua
mixture. These expressions were obtained from the equations of motion,

Eq. (12) of 3.2,by neglecting the acceleration terms (quasi—static assumption)
and setting the body forces equal to zero. Equation (2) is the constitutive equa-
tion for the solid partial stress tensor 0', j . Equation (3) defines the partial
fluid stress tensor. Equation (4) gives a constitutive relationship for the dif-
fusive resistance in the media 1r; . (Diffusive resistance results because
interaction occurs between the two component materials when the continua are

deformed.) Finally, Eq. (5) is a constraint that is imposed on the system of

 

The subscript It preceded by the comma denotes partial differentiation
with respect to x k .

28

29
equations because the fluid that fills the pores is incompressible.

The system of Eqs. (1) through (5) contains 29 unknowns and consists
of 19 equations. However, if the six solid strain definitions, the three dif-
fusive force definitions, and the definitions of the three normal components
of deviator tensor fij are introduced, the system is complete; containing 31
unknowns and 31 equations. The purpose of this section is to reduce this sys-
tem of equations to a four-by-four system of equations with the solid displace-

ments 0; and the fluid pressure p beingthe unknowns. Adding Eqs. (1) and (2) gives
(Tin t ”in ' 0) P 3i): r P 5m t 202 °ik * °3°mm 5m (6)
By definition, 11') is expressed as follows [4]:
I
77‘ 35(oki' ”HI,“ (7)
if the body forces are zero and the deformation is quasi-static. Substituting

this expression in Eq. (4) yields Eq. (8).

1 air av-
iiaki‘fiki),k'0(a_'."a—t') (8)

If the equilibrium equation (1) is divided by two and subtracted from Eq. (8)

the following expression results:

so; dvi
’"kiflt'aié—g'a—g’ (9)

The substitution of expression (3) for 1rki in Eq. (9) yields Eq. (10).
air av
p,, . ots-i'a—gi (10)
This expression is of the same form as the modified Darcy law proposed by
Biot [4] to describe the creeping flow of a fluid through a deforming porous

media.5

 

The microscopic flow through the pores is viscous and temperature
dependent. The viscous effects are accounted for in coefficient

of Eq. (9), which relates the macroscopic fluid discharge through the
media to the macroscopic pressure gradient 9.; . In order to ac-
count for the temperature effect, it would be necessary to consider the
coefficient to be temperature dependent. In the present problem, how-

ever, 0 may be considered constant because the deformation is as-
sumed to occur at a constant temperature.

30
Substituting constitutive equations (6) into the equilibrium equations (1)

yields Eq. (11):

2°2°ik,k*°3°mm,i"('*°I)9:i ' 0 (11)
The artificial solid strain tensor is defined as shown in Eq. (12):
'ik'é’ium 4' “k.i’ (12)
Taking the divergence of the solid strain tensor Eq. (12) gives Eq. (13):
05m ' "gt Us,“ + “u“ (13)
Substituting Eq. (13) in Eq. (11) yields Eq. (14):
020i,kk+i02*°3l¢mm,i"(OIHHM'O (14)

Using vector notation Eq. (14) can be expressed as shown in Eq. (15):
o2v26+(o2+o3fi($-m—(o. “fin-o (15)

Equation (15) is analogous to the Navier displacement vector equation of
elasticity. Because of the presence of four unknowns “i and p in the three
Eqs. (14), another equation is necessary to provide a complete system. A

fourth equation can be obtained taking the divergence of both sides of Eq. (10).

e
Vzp - at STm-fmm) (16)

can be eliminated from Eq. (16) by substituting expression (5). Mak-

 

Imm
ing such a substitution and utilizing the definition of a, , given in Eq. (26) of
3. 3 results in Eq. (17):

v29 . 0:? gfitmm (17)
In order to eliminate the dependent variable 9 from Eq. (17), another expres-

 

sion for Vzp can be obtained from Eq. (15) by taking the divergence of that

equation.

(202 +03)

‘72,. _——_m H) v am, (18)

Substituting expression (18) into Eq. (17) eliminates p as shown in Eq. (19).

oto,+|) 1 “mm .1. 60mm
v23mm ' (202+03l (f) at I c 6' (19)

Equations (15) and (19) are the fundamental equations that govern the

deformation of a porous linear elastic solid saturated with an inviscid,

31
incompressible fluid. The solution of Eqs. (15) and (19) in conjunction with suf-

ficient boundary conditions will provide the solutions of the quasi-static prob-
lems of poro-elasticity.° In the next section, the solution of Eqs. (15) and (19)
through the use of displacement generating functions is discussed for the case

of axially symmetric deformation.

4.2 Axially Symmetric Deformation - Displacement Generating Functions
The deformations of porous media considered in this study are limited
to the special case where the displacements are axially symmetric. In this
case, it is possible to further simplify the defining differential equation. In
this section, two unknown functions Eir,z ,t ) and S (r,z,t) will be intro-
duced from which the displacements in the interacting media can be computed.
In order that the defining differential equations be satisfied, it is necessary
these functions satisfy certain differential equations; however, the equations
that must be satisfied are much simpler than those that govern "i and p .
Equation (20) expresses the axisymmetric constraint on the solid dis-

placement vector.
G-u,(r,zl§, +uztr,zléz (20)

The Laplace differential operators 6 and v2 which appear in the

fundamental equations (15) and (19) have the form of Eq. (21).

23311'32 23* .e. la a 21
‘7 'ar2+rar+7é§75+azz'V"'ar+‘orao+'zaz ‘ ’

in cylindrical co-ordinates where 3, , 3, and 82 are unit vectors tangent

to the coordinate curves.

 

6 Alternatively, if ‘mm is considered as a fifth unknown, in addition to

iii and P, then Eqs. (15), (18), and (19) may be considered the basic
governing equations.

32

Assuming a displacement field defined in terms of two unknown functions,

E and S as shown in Eq. (22)

Ui'c1.l)'$(E‘(f.l,t) +zs (r,z,t'))-2$ (mun: (22)
where

v28(r.z,t)'0 (23)
leads to solutions of Eqs. (15) and (19). if E is properly defined.

Taking the divergence of both sides of Eq. (22) shows that the dilatation that

results from assumed displacement field depends on E only.
3°U=0mm'v28 (24)
Substituting Eq. (24) into Eq. (19) of 4. 1 yields the partial differential equation

- which E must satisfy.

4.12. 2
‘75 catvE as)"

If Eq. (22) is substituted into Eq. (15) of 4. 1, which must be satisfied also, an

expression for 3,, results.

__ _. (2024-03) 2 + 202 <95

w" v (I+a.) (Hon) 3; (26)
Solving partial differential equation (26) yields Eq. (27) for P .
o 20 20
-(i_2_)v2 +___2_..£+po(n (27)

' (I+o.) (I+o.) dz
where PO“) is arbitrary. The physics of the problem being studied here

insures that 90‘” a o . That is, the initial fluid pressure at time t a 0’ is
assumed equal to zero, and no uniform hydrostatic pressure is applied to the

fluid continuum at any time t a O . Therefore, Eq. (27) reduces to Eq. (28).

:(°3+2°2) v2 + 2°_2 fi (28)
. (I-t-a') “+09 62

In considering the axisymmetric deformation of materials with pore
compressibility R equal to unity, McNamee and Gibson [_§] proposed

two displacement generating functions that satisfied equations similar
to Eqs. (23) and (25).

 

33
The displacement field defined by Eq. (22) may be expressed in physical com-

ponent form as shown in Eq. (29).

fir-g? +z§-§,uz-§E+z§%—s (29)

Partial differential equations (23) and (25) are the equations that define
E and S and thus the axisymmetric deformation of the poro-elastic founda-
tion. In order to establish the boundary conditions on these equations, it is
necessary to consider the deformation that occurs in the plate that is sup-
ported by the foundation. Therefore, the equation that governs the plate defor-
mation must now be considered. First, however, a few comments concerning
the arbitrariness of E and S are given in 4. 3.

4. 3 Arbitrariness of E and 8 Functions

 

The functions E and S are arbitrary to the extent that the partial stress
tensors are not affected if an arbitrary linear function is added to E or if a
constant plus a term linear in r is added to S . If E is modified as shown

in Eq. (30)
8':- E +b. +152 r + b3: (30)
the rigid body displacements are shown in Eq. (31).
u', I “r 4- b2 , u'z' uz+b3 (31)
It is observed that the addition of the constant b. does not affect the displace-—

ments. If S is modified by adding a constant b4 as shown in Eq. (32),

s‘ = s + b4 (32)
the rigid body displacement (33) becomes:

W2 8 uz +b4 (33)

Further, if a linear function in r is added to S as shown in Eq. (34) :
-, J. 32.2: . 34
s S+b5r.2(az ar) 2b5 ()

a rigid body rotation results.

34
It is interesting to note that the addition to S of a linear term in 2 does

not affect the artificial solid displacements, but does cause a change in the

partial stress components on the artificial fluid continuum as shown in Eq. (35).
S. . S+b52, u'r'Ur, ".2 .uz

(35)

20 b

-‘]f"_" I -759 8 -72.! s p': +45.

V

THE VISCOE LASTIC MATERIAL

In order to define the deformation of the plate, it is necessary to relate
the strains occurring at any arbitrary point in the plate to the state of stress
at that point. It is the purpose of this Chapter to set down such constitutive
equations for the viscoelastic plate material. It is shown that a single time
operator may be used to define the constitution behavior of an incompres-
sible linear viscoelastic material, and that a hereditary integral may be used
if the deformation is quasi-static.

The constitutive equations of the ideal, linear, isotropic, viscoelastic
solid exhibit a time-dependent relationship between stress and strain. A
convenient form of the viscoelastic constitutive equations results if the dilata-
tion is related to the first invariant of the stress tensor and the deviatoric

strain is related to the deviatoric stress tensor as shown in Eq. (1).

RH)

0mm " 3 n-2 (m cm
o..=_sm__ (1)
O" u+v<m 5'1

where

R (l) and V (t) are linear time-dependent operators which are analogeous

35

36

to the Youngs modulus and Poisson's ratio constants that appear in con—
stitutive equations of ideal elasticity. In this study,it is assumed that the

viscoelastic plate material is incompressible and, therefore, the operator

V“) must be constant and equal to 1/2 as shown in liq. (2).

3(I-2 v) ‘
Emm:-——————— 0mm: 0 (3)
R0)
This constraint on V eliminates the first constitutive equation (1) and sim-

plifies the others to the form Eq. (3).
This expression shows that one linear time operator defines the constitutive
behavior, if the viscoelastic material is incompressible. Using the deviatoric

tensor definitions (1), it is possible to express Eq. (3) in terms of the total

stress and small strain tensorial components.

2 3 .. - __'_ ..
5n W i an 2mm 0mm in (4)
In the case of uniaxial stress, Eq. (4) reduces to Eq. (5):
Gill) ‘ RU) Em“, (no sum on j ) (5)

which shows that a uniaxial experiment can be used to derive R“)

A hereditary integral type constitutive equation may be used to describe
the viscoelastic plate material in uniaxial strain if the stress relaxation be—
havior of the material is known. The necessary data are obtained by sub—
jecting the material to a constant uniaxial strain and observing the stress
that results as a function of time. The stress relaxation function 4’“) is

defined in terms of the uniaxial test data as shown in Eq. (6):

CTN?)
¢(i) ‘51:?» (no sum on j ) (6)

where the instant of load application is defined as time zero.

37
The general form of a typical ¢(1) function is shown in Figure 2.

In order to approximate the stress response for a more complicated
strain loading history, the linearity of the material can be utilized. This is
done by superimposing the stress relaxation response to several step function
loads such as shown in Figure 3, where the actual loading curve 8‘ j DH) is
approximated by either of the two step function curves shown. The stress
response to each of the approximate loading curves is given by Eq. (7).

0’“ a §¢u("lk)%t? Atk 3 Gun“)
A81 (7)
a" a §¢l("'k) 2—,; Atkz 0"”)0)
Letting Atk approach zero in both summations (3'u and 02 yields Eq. (8):
inf/1mm (7,.I 8 supromum oz I/Mt-‘l‘li‘jezg-(Bdf (8)

'o
where the last equality follows from the definition of the Riemann Integral.

Letting lo approach -co yields an expression for 0 (t) account—
ing for the effect of the entire uniaxial strain history of material. Inte-
grating the resulting expression by parts and assuming that the material was
unstrained at c s «0 gives Eq. (9).

t
d ( m a t
/¢(i-T)—d-¥— dt ' NO) Sun“) +/€(m(‘n a (if (9)
For the problem being considered it is assumed the loading history of the

material prior to application of the strain loading at time zero may be neg-

lected. This assumption leads to the hereditary integral constitutive equation

(10) for uniaxial strain.

00M" - macaw) +Zew<ni¢l§€flfl d1 (10)

Eq. (10) is equivalent to the operator form (11).

STRESS RELAXATION FUNCTION

38

 

 

 

 

TIME

Figure 2. Typical stress relaxation function.

 

 

 

I

o

C

C
-—_———_~d-
_————_————-L
—-———————

 

L—io _. L—Ai—J

 
     

————_—_-_

V

 

Figure 3. Multi step-function approximations of strain loading.

 

 

3f)

amt-T)
0.”,(0 - [MOM ) + c.,/'( l—fi— d7] 5.1M” (11)

Comparing Eqs. (5) and (11) shows that RU) may be defined by Eq. (12),
R(t)= ¢(o)()+6ft()%'T—41d‘r (12)

if the deformation is quasi-static. Substituting Eq. (12) in Eq. (4) yields the

form of constitutive equations that are used to describe the viscoelastic plate

material in this study

,, .. ecu—1') , §_ _
¢(O) 8”“) +/€U(fl—éi..— dT 2 O"

CJ'rnm
-_ . 13
'l 2 5 ( )
o

Ii

VI

VISCOE LASTIC THIN PLATE THEORY

6. 1 The Viscoelastic Plate Deflection Equation

 

In order to define the deformation response of the plate as a whole to
the externally applied load and the foundation reaction, it is necessary to insure
point-to-point equilibrium throughout the plate . The equilibrium equations for

the viscoelastic continuum are given by Eq. (1):
oij,i '0 (1)

where 0' i j is the stress tensor.

The constitutive equations (13) from Chapter V, equilibrium equations (1),
and small strain definitions can be used to derive a viscoelastic plate bending
equation which is analogous to the differential equation that describes the bend-
ing deformation of thin, perfectly elastic plates. In the derivation that follows it
is assumed that:

1) the plate surfaces are subjected to normal stress tractions only

2) there are no forces applied in the plane of the plate

3) the body forces are negligible

4) the vertical plate deflection w is small in comparison to the plate
thickness

5) the plate thickness is small in comparison to its in-plane dimensions.

40

4i
Restrictions 1) through 5) above, lead to the following assumptions con-

cerning the form of the plate displacement field '3 .

1) Lines normal to the mid-surface of the plate are assumed to remain
normal to the mid-surface during the deformation (shearing deformation is
neglected).

2) The mid-surface of the plate is assumed to remain unstrained during
the deformation.

3) It is assumed that the normal stresses perpendicular to the plate sur-
face do not significantly affect the strains in the plane of the plate.

These assumptions yield displacement component functions of the form (2):

a a
us-za—‘g-JI-zfi (2)
where u and v are the displacements in the plane of the plate and

w is the vertical displacement component. The strain components can be
computed using Eq. (2), as shown in Eq. (3):

c 32 6‘
Saul-'13:} 'eyv"‘5'9!‘ ' 6W "13%; (3)

Holding time 1’ fixed, but arbitrary, and integrating the equilib-

rium equations (1) over the thickness of the plate yields three resultant
equilibrium conditions which must be satisfied. Two of the resultant equa-

tions are automatically satisfied, the third is given by Eq. (4):
6 K g! 4'- "' s
it“ + ay 4- q q o (4)

where Q,‘ and Q, are the resultants of the 0;“ and 0'” stresses, and
(1+ and q" are the stress tractions applied to the upper and lower faces of
the plate. Two additional resultant equations can be obtained by computing the

moments of the first two equilibrium equations about the middle plane of the

plate, and then computing ,the resultants of these equations over the depth of the

plate. The resultant moment expressions are given in Eq. (5):

42

6M 6M
_l‘_ __!Y .. .
ax ay 0,, 0
6M all (5)
- _.LV _J. - :
ax 6y 0! 0
where h/2 h/Z h/Z
M" l/O’yxzdz , M‘s /O“zdz , M’s/Oyyzdz
-h/2 -h/2 -h/2

As in the case of thin elastic plate theory, displacement solutions of the form
Eq. (2) are sought such that equilibrium is satisfied in the resultant sense as
defined by Eqs. (4), (5), at all times i . Eliminating Ox and Qy from
Eqs. (4) and (5) yields Eq. (6):

m_2_"fflr+flx_.q—-q+ (6)
av axay ay‘

This single differential equation must be satisfied to insure equilibrium in the

resultant sense.

Equation (6) can be expressed in terms of the vertical displacement com-
ponent.w.by substituting the strain equations (3) into the constitutive equations

(13) of Chapter V and computing M, , M,‘y , and M from the resulting

Y
expressions. In doing this, assumption 3) is used to simplify the constitutive
relationships, and 1’ is held fixed, but arbitrary, for these operations. Sub-
stituting the resultant expressions into Eq. (6) yields the thin plate equation of
viscoelasticity in Cartesian co-ordinates.

Using symbolic notation the plate equation takes the form Eq. (8):

DMV’V’VI - q” «1'. cm I 5&9 h’ 5 DR (t) (8)
in any admissable co-ordinate system. .

43
6. 2 Solution of Plate Equation

 

In this section, an iterated Laplace-Hankel transformation is used to solve
for the transform image of the plate deflection in terms of the transformed
foundation reaction. This provides a boundary condition for the poro-elastic
foundation boundary value problem. With this expression, sufficient boundary
conditions are known to define the foundation boundary value problem.

Considering the mathematical model that is being analyzed, Eq. (8) is
restricted to a cylindrical co-ordinate system, and w is assumed to be a
function of radius r and time 1 only. Figure 4 illustrates the geometry of
the plate, the reference co-ordinate system and the surface traction sign con-
vention. The 2 axis in Figure 4 extends downward into the half-space foun-
dation.

The plate equation (8) takes the form Eq. (9):

owe—a,“ r 37x37: +73)w(r.t t) a q+(r, n-q- (r t) (9)
where, ‘11,.” . {HUN . r< b
0 ,r>b
(10)
and I, 120
H“) ' {o,t<o
(11)

in the cylindrical co-ordinate system for the axially symmetric loading shown
in Figure 4. The bending stiffness om , defined in Eq. (8) , contains
the time operator Rt!) from Eq. (12) of Chapter V.

There are two unknowns in Eq. (9); the plate deflection w and the reactive
pressure q" on the lower face of the plate. An additional equation relating

these two unknowns will be obtained subsequently by considering the defama-
tion response of the half-space of poro-elastic material which supports the

44

.033 came—ocean? o5.

. v 953..”

 

 

 

 

 

 

M s.
35...
n n n n n n n n
i... v

0. (13035 >432...
5 2e...

 

 

 

 

loaded plate.
It is useful to modify Eq. (9) so that the partial differentials in r can be
eliminated [2, lg] . This can be done by utilizing the zero order Hankel inte-

gral transformation that is defined by Eq. (12):
co

to (q) téfrflr) Jo (qudr (12)
Applying this transformation to both sides of Eq. (9) and rearranging terms
yields Eq. (13):

t -
D [Mal wo (mt) +{wo(q,tl gig-Fa d7]

(13)
. {5 ha, (an) H (t) +q;(n,t) m‘

for the transformed displacement wo .
Further simplification can be achieved by applying the Laplace integral trans-

formation [§_, lg] , defined by Eq. (14):

it.) few)... (14)
to Eq. (13). The final result is given in Eq. (15):
- - bJ(b ) Elms)
sD¢(S)w°(q,s)-q—'-‘lql +-°—n,— (15)

which is an algebraic equation in q and 3 relating W0 and a; . Solving
Eq. (15) for 63 yields Eq. (16):

a; s q“: 0W3) 17001.8) - (ugly! (16)
which may be regarded as a boundary condition on the foundation at the
plate - foundation interface. It is noted that although the plate equation has
been reduced to an algebraic equation that relates a; and {8° , neither of
the unknowns a; and We can be determined explicitly from this single equa-

tion. It is, therefore, necessary to investigate the foundation-plate interaction

in order to determine a; and We

VII

THE PLATE ON THE HALF-SPACE

There are two unknowns in Eq. (16) of 6.2, the transformed plate deflec-
tion ‘0 and the transformed reactive pressure a: . An additional equation

relating these two unknowns can be obtained by considering the response of

the foundation below the plate. This can be done by formulating an

initial-boundary value problem for the foundation. It is convenient to

use a cylindrical co-ordinate system because the loading and deformation

are axially symmetric.

7. 1 Boundary Conditions

The constitutive equations for the solid partial stress tensor, given in

Eq. (37) of 3.4,can be expressed in cylindrical co-ordinates in terms of phys-

ical components as shown in Eq. (1):
an do u an
0'":- o.p+202—-Lar +03( or + , +_az )

u an u au
°oa"°nP+202 ++°3ifi+4+4

" dz

)

a a (1)
uz “r
or + 62 )

 

0'2 ' 02‘

all; ally Ur all:
011 “dip-#202 <32 +03( <3! +—+ )

r 62
In order to determine the boundary conditions on E and S, it is necessary

to express the stresses in terms of E and S . Substituting Eqs. (24), (28),

46

47

and (29) of 4.2 in Eq. (1) yields expressions for the stress components in terms

of functions E and S .

o -2 2 2
o":( 3 0:02)v2£+202(a 5+: a s_ a. as

 

 

n+0.) 6,2 arz (Hall-52—
0,; =202(g-;25§z+z 3726—52) 0'0. 020:0

The boundary conditions on E and S can be determined by considering the
state of stress at the boundaries. At 2s 0 , the total stress in the z direction
on the 2 face of an element of the poro-elastic material must equal at the
plate reaction. This means that the traction stress q+ is exerted on both the
fluid and solid components of the solid-fluid mixture. Using the definitions of
4. 2, this boundary condition can be expressed in terms of E and S as shown in

Eq. (3).
2
ON m) - 202 (355 -33- + VZE)

It is assumed that the traction q+ at the surface of the half-space van-

 

, . 0 "° (3)
ishes as r becomes large. This is physically apparent since q+ is the re-
action that results under the plate when a circular uniform load of finite radius
is applied to the top surface of the plate.
The surface at z . o is assumed to be impermeable to the fluid at

any time i . Therefore, the relative velocity vector of the fluid to the solid
material must be zero in the z direction at the surface. Equation (10) of 4. 1
states that the partial derivative of P with respect to z is proportional to the

relative velocity in the z direction and, therefore, is zero at z a o . This

48

condition is expressed by Eq. (4).

g 625 ("3+202’ v2a__E_ 3 t>0 4
Ho. 32-2 20 az) 2800 '20 ()

The shear traction on at 1.0 is equal to zero, as it is assumed that
the plate is free to deform horizontally relative to the half-space surface.
This condition is expressed by Eq. (5):

a2E
202(—- z-tur,,ot)) -o i>0.r20 (5)
It is assumed that the vertical displacement in the foundation at Z = 0

is equal to the plate deflection, as shown in Eq. (6).
u ( no.1) 8 Mr,” (6)

It is also assumed that the stresses, and displacements, and rigid rota-
tion in the half-space vanish as 1'2 + :2 approaches infinity. These con-

ditions are expressed in the following equations:
mu 0'” 8 0 lim 0}: 8 0 lim 0'" 8 O

r2+zz-b¢o

(7)

lim p80 lim 012:0
limu, 80 lim u2 = O
.§_§.

lama, 0

By considering Eq. (2) of this section and Eqs. (24), (28), and (29) of 4.2, it
can be seen that the above physical conditions are satisfied if the following

restrictions are placed on E and S :

2 <92

. 2 . . Q5, 66 S
llmV E 0 "ma: 0 "fills—'22 + Z :22) 0
Jf2+12—D

aE a_s a?- ___E_a azs .

62 a2 (8)

a_§ e__s a E s .

liml— +62 -Sl8o lim irrdz‘o'zdrdz’ 0

lint-37580

49
When the step load is applied at time i=0 , the plate will re-

spond with an instantaneous elastic deformation, and the skeletal structure of
the porous media foundation will undergo an instantaneous volume change be-
cause the skeletal material is assumed to be compressible. This means that
at time zero gmm takes on an initial value which places a restriction on the

function E(r,z ,t ) as shown in Eq. (9).
am (r,z,o) :sz (r,z,o)8 flr,z) (9)

This is the initial condition that is imposed on Eq. (25) of 4.2. The

function , H r, I) ,can be determined by considering an elastic plate, with a
Youngs modulus equal to R (o) . supported on an elastic half-space that has
Lame’constants 02 and (04 42:5 “8) . The initial condition is developed
explicitly in 7.2 and is stated here for reference:

2 l °° -
v E(r,z,o)8 C '12.] (b N d 10)
(a4-‘2’fiae4-oz) of qe ' q om!) q (
where: ,
202(04-‘2508+02)b

DR‘O) (04-525084-202fll3+202(a4—(z)508+02) (11)

 

7. 2 Initial Condition on v25

 

In order to solve for VZE 8 9mm in partial differential equation (25) of
4.2, it is necessary to know the initial value of 3mm . At i=0 the con-
stitutive equations for the partial solid and fluid stresses reduce to Eq. (38) of
3. 4 which may be added together to give a relationship for the total stresses

as shown in Eq. (12).
03] ”’ij ' 202%] +(a4-‘3‘baalemm 5 ii (12)
In addition to satisfying the constitutive equations (12), O’U- + 11'; j must also

satisfy the equilibrium equations (13) of 3.2 at time zero.
“’33 *Wijm '0 (13)
Therefore, determining the initial condition on VZE 8 9mm reduces to finding

the solution to the analogous 'elasticity' problem defined by the total stress

50
tensor constitutive equations (12), the equilibrium equations (13), the Beltrami-

Michell compatibility conditions [il] , and appropriate boundary conditions.
The boundary conditions on on +17” can be determined by considering

the magnitude of the stresses at time zero, at the surface of the half-space.

The total normal stress applied to the top surface of poro—elastic half-space

at time zero is q*(r, o) as expressed in Eq. (14).
“22 (no.0) *"zz(r.0.o)-q(r,o)+ (14)

According to the assumed frictionless boundary condition at the plate - half-

space interface, the shear stresses vanish there.
0}z(r,0.o) +1r,z(r,o,o) =0 (15)

In order to obtain an expression for q+( r, o) , the Hankel transformed
plate equation (13) of 6. 2 can be considered in the following modified form,
which results when the time t is set equal to zero.

DRtolwotq,o)- 303 bd. (b q) + 53—333 (16)
The unknown transformed plate deflection "o in Eq. (16) is assumed to be
equal to the transformed vertical deflection of the half-space at z . o .

This condition is stated in Eq. (17).
Wain.” ' “20mm.” (17)
The deflection uz (no) resulting from the axially symmetric normal

surface traction at z 8 O on the half-space can be computed using the

solution of Terrazawa [g] as shown in Eq. (18).

 

0 tr: 0) ._ ‘a4‘u’5aai-azl ”thlgntqng (18)
where l ’ ' 1'0 202(04-‘2’508+02)[ fl

2
"((132 ' q; “l: O ’
Taking the zero order Hankel transform of Eq. (18) and substituting for the

transformed plate deflection "o in Eq. (16), results in Eq. (19) for q; at

time zero.

DRlole 04-‘2'5034- 202) z (q)_qu. ( on)
Zozt aimfiaa + 02) '1

 

03 ( q,o)-- (19)

An explicit expression of 201) is desired since 0' mm i‘ ”mm can be com-
puted [1_2_]if Zln) is known. Substituting the second equality (18) in (19) and

solving for 2M) provides the desired result:
Z(q)=CqJ|(bq) (20)

Where C = ‘202(a4"2508+02)b

 

DR(Olq3(a4'(2)508+202)+202(04J2508+02)
Finally, using the Terrazawa solution again, with Eq. (20) substituted for 2(a).

the solution of 0 mm +1rmm is obtained.

3a -3‘2’fia +2o ) 0° -
-‘ 4...- 8 2 / Codiibnie “’Joinndn (21>
i=0 (“4" Pas+°2) 0
By examining the constitutive equations (12) it is apparent that emm(f.z.o)

 

 

(omm-t-nmm)

 

is related to Omm‘H’mm by Eq. (22).

(Omm'l'fimm) =(2°2+304-3‘2508)9mm(rtzto) (22)

i=0
Equating Eq. (21) and Eq. (22) and solving for 9mm (7,2,0) yields the

 

desired initial condition as given by Eq. (23):

0°C ’“zub )J( )d
v2E(r,z,o)=omm(r,z,o)8/ qo m: '1 our :1

 

(23)

If this equation is operated on by the zero order Hankel transform, con-

dition (24) results.

i qu’flzilibq)
(04—‘2’54'02l '1

 

°mm° (n.2.o)= (24)
The transformed quantity °mmo('l.1.0) provides an initial condition in
mathematical terms accounting for the instantaneous volume change that occurs

in the foundation upon application of external loading. This condition is nec-

essary to the solution of the partial differential equation that defines E .

52
7. 3 The Initial-Boundary Value Problem
The results obtained in 7. 1 and 7. 2 can be summarized by outlining the
initial-boundary value problem. Two partial differential equations must

be solved:

cv4E(r,z,:) av? 3% (r,z,i)

Vz‘Slr,z,i) . o (25)

where 1’0. '30.l>0

The solution of the first equation in (25) will be required to satisfy the fol-

lowing initial condition:
V2Etr,z,o)8f(r,z) (26)
The solutions E and S must also satisfy boundary conditions (27) through

(29) at 2-0 .

a_?-__E

a -—62 ”MW 80 r20,t>o (27)
a2 as

2"2 (555‘s: + VZE) 8 (I‘M?) r20, 180, t>0 (28)

2 (0+20)
as+ 3 zvzag

5:2 752— .. '0 '20:“0» ”0 <29>

The solutions will be required to satisfy boundary condition (30) also.
, 0 (non) - vim) (30)
As 72 + :2 approaches infinity (31) must be satisfied.

. a 62E azs , aE as
umsz-o Innis-$80 'iM(§_-22+z:z—2’ 0 lim(— +237) -0
“'2'? 2-009
a2E azs aE a__s_ a2E azs (31)
limta jam—zho "mtg-1+ 3; -S)8o 'imlcT—raz+25w_z'l'°

"mg-$-

r O

53

A method will be developed for solving the system of Eqs. (25) through
(31) in Chapter VIII. Generally, the method will consist of applying integral
transformations to the problem in order to reduce partial equations (12) to
ordinary differential equations having one independent variable 1 . Trans-
formed initial and boundary conditions will then be imposed and transformed

solutions for the stresses and displacements will be established.

VIII

SOLUTION or INITIAL-BOUNDARY VALUE PROBLEM

The problem has now been defined and reduced to terms that allow solu-
tion. The initial-boundary value problem that defines the foundation defor-
mation is summarized in 7. 3. In 8. 1, the governing partial differential equa-
tions will be solved using iterated Laplace-Hankel transformations. In 8. 2,
boundary conditions will be transformed and imposed on the solutions for E0
and §o obtained in 8. 1 and transformed solutions for the plate deflection,
foundation reaction, and several other variables will be obtained. Finally,
in 8. 3 the transformed solutions of the remaining unknowns are obtained.

8. 1 Solution of Differential Equations Usigg Transform Methods

The first differential equation (25) of 7. 3 takes the form (1):

60mm
1
a? ( )

when sz is replaced by 'mm . Taking the Laplace transformation of Eq. (1)

 

cvzomm 8

eliminates the partial derivative with respect to t and reduces (1) to a partial

differential equation in r and z .
cvzimm(r,z,s)=83mm(r,z,s)-omm(r.l.0) (2)
Equation (3) is the Hankel transform of (2):

- .. o ,
(ia‘nzhmmoima)=%°mmo(n.z.s)--1cw (3)

which contains derivatives with respect to 2 only. Substituting the initial con-
dition expression previously obtained for ommo(q,z,o) in 7.2 results in

Eq. (4).

54

55

d2
(-—-(n2+-))immo = ko" “2
dz2 (4)

l I
where k 8 - Cq -— J| (M)
(2)

The general solution of the ordinary differential equation (4) is given by Eq.(5):

ammoinflfil'me'nz+3204” *33°+”z (5)
where .9 k ’ k
B|a-ks ,32=d|+m,83ad|+m.

d| and di are arbitrary constants
and II 3 VH2 4' "c

From the definition of the displacement function E (r, z , t) it has been shown
that 'e'mmolq ,2, a )may be expressed by Eq. (6):

(dz-Zlit )8‘ ( )

52 'l o q,z,s °mmo 'lvzvs (6)
Substituting Eq. (6) in Eq. (5) yields the differential equation that defines
Eo(q.z.s) .

d2 2 - -qz -Ilz +112
(az—é-q )Eo(q,z,s)8B,e +82e +834: (7)

Equation (7 ) is an ordinary, non-homogeneous differential equation and its

general solution is given below:
Eolnozos) 3 (Cl + d2, 3.“: +(C2 + d'Z) 9'12 + Ca 2"“ (8)
+ (2419“2 + C5 0""

where
B B 8
Cl 3 (-.—'. + ——2__ .. .—3_)
4:12 aqtu-n) zqtu+ni
C28 (c-B— — —§2_.+ 8
4:12 2q(u- n) qu-a)
. Jen. L3° .32°
C3 2:13 ' C4 ' C5 T

and 42 , dé are arbitrary functions of q and a .

56
The second Eq. (25) of 7. 3 is the governing differential equation for the
displacement generating function 8 . Applying the Laplace and zero order
Hankel transformations to that equation gives Eq. (9):
.23. _ 2 - .
dz? :1 ) So(q.z.s) 0 (9)
which is a homogeneous ordinary differential equation in z . The general solu-

tion to this equation is given by Eq. (10):
go (q,z,s) 8 d3e"lz + dgeqz (10)
where 63 and of, are arbitrary functions of q and s .

This completes the solution of the differential equations that define E0
and So . Equation (8) is the solution for E0 , and Eq. (10) is the solution
for $50 . The unknown coefficients multiplying each term in Eqs. (8) and
(10) are constant with respect to z , but must be regarded as functions
of q and I . In 8.2 the transformed boundary conditions will be
imposed on the solutions E0 and §° , and this will result in the unknown
coefficient functions becoming fully defined.

8.2 DeterminingfifLaplace-Hankel Transforms

If the first three boundary conditions at z = co given in Eq. (31) in 7. 3
are now invoked: the general solutions E0 and §° can be simplified. The
first of the conditions implies that ammo must approach zero as 2 ap-
proaches infinity. Therefore, the coefficient 83 in Eq. (5) must vanish.
According to the definition of , c4 given in Eq. (8), C4 must also vanish.

83 8 C4 = O (11)

The second bomdary condition at z a co implies that 339- must vanish as
Z approaches infinity. This condition makes it necessary that d3 be equal

to zero as shown in Eq. (12):

57

lim 5-5-0 - a z-
2*m dz (n.2,3)’nd3 9'1 -0 (12)

 

Therefore, Eq. (10) reduces to Eq. (13):
§o (n,z,s)= d30'"z (13)

The constraint on the transformed quantities that results from the third con-

dition is given in Eq. (14) .

2" 2’
lim (6 E0 +2 a So
z“°° 622 622

Substituting from Eqs. (8) and (13) and invoking Eq. (11) results in Eq. (15):

 

 

i=0 (14)

21'0"; (q2(02+d'2) 9"“ 2c325'" +n2 d35"z)=0 (15)
For Eq. (15) to be true, the constants Ca and d'z must satisfy condition

‘16" topaz, )-o

(16)
Now consider the boundary conditions at 1'0 . elf Eqs. (27) and (29) of 7. 3

are Operated on by the repeated Laplace-Hankel transformations, the two

conditions given in Eq. (17) result.

déo
'n— dz aims, si- 0
2g (17)

_£ , ,3 1’ =
(I! O ) r222- '12 '—;',(n,o 8)
Substituting for 6E0 in the first Eq. (17) results in the following expression:

(18)
“(0' +62)'C3‘ "05

An expression can be obtained from the second Eq. (17) by substituting for

bOth E0 and so 0
(0 +20 ) 2 2
. - 2 C + -M C
as 41—4202“ t :12 3 um ) 5) (19)
Taking the Laplace-Hankel transformation of boundary condition (28) in 7. 3

yields Eq. (20):

 

3 The remaining conditions (31) of 7. 3 are satisfied.

dS - ..
-2¢2(d—z9(q,o,s)-q2 Ea (q,o,s))' a; (20)

Substituting for E0 and So gives Eq. (21).
- - - 2 . -
202i “('3 II “C' +d2 )+C§ H G; (21)
The Laplace-Hankel transformed plate equation (15) of Chapter V1 is re-

peated here as Eq. (22).

.4-
Woln,s)Ds$(s)=:?sbJI(bq)+ (10:2,) (22)

 

The plate deflection is assumed to equal the half-space deflection at z = O for
all no . -Therefore, 970 can be related to E0 and §o as follows, using

Eq. (29) of 4.2:

_ _ aE -
wo(q,s)=uzo(q,o,s)=fi‘mofi)‘So('lv°»5) (23)

It then follows by substitution for E0 and 's'o from Eqs. (8) and (13) that

E . 4 is alid.
q (2) v Wo(q,s)'-Q(C.+d2)+C3-u05'd3
(24)

Substituting Eq. (24) into the plate equations (22) and making use of Eq. (18)

 

yields Eq. (25). + b.) (b )
.. =_ 4D — d _q I '1
Go q s¢>(s) 3 —— (25)
Solving Eqs. (18), (19), and (21) simultaneously yields an expression for d3 .
.+
cs-QO (26)
d38———C
where 7
C . k02(q+u)
mum?
and
7 a (202)2 n3-202q2fl(03+202)-202 qu2(o3+202)
”(n+qH03-6-202)

Substitution of this result in (26) yields the following transformed solution for

the reactive pressure a: under the plate.

3'; . 15:2 Dames-07m, (bn)
we, - q4085(8))

Substitution of expression (18) for q(C,+d2) in Eq. (24) simplifies the

(2 7)

59

transformed plate deflection expression to the following.
3001.3) --d3 (28)
Substituting the expression obtained for da into Eq. (28) yields the trans-

formed solution for the plate deflection. -,

 

 

_ Q 'C
W°(ng.) . OCT 6 (29)
The solution §o follows directly from Eq. (13):
- (Cs-63) - z
so‘flazos)= C an (30)

Now referring to Eq. (8) for E0 , the constant C|+d2 is related to the
known constants C3 and C5 through Eq. (18). Also, (02+d2) and C4 are
equal to zero by Eqs. (16) and (11), respectively. Therefore, in order to
define E0 completely,it is only necessary to determine C5 . Equation (21)
can now be used to solve for 05 since a; is known by Eq. (27). Substituting

for (CI‘HiZ) from Eq. (18) into (21) and solving for 05 results in the fol—

 

.lowing expression: ' 53-202“ (d3+c3)
C = l
5 -
202a (q u) (31)
E0 then is given by Eq. (32), with 05 being defined by Eq. (31).
E ( .z,s)-(-"— £-£c )5"! La -nz ~11:
o 'l 2q2 8 q 5 +2'l s 20 +C5e (32)

Some of the transformed stresses, strains, and displacements in the
half-space foundation can now be computed using the solutions (30) and (32)
for §0 and ED . 310 may be obtained by transforming the second
Eq. (29) of 4.2 and substituting for E0 and So .

(CS'a3)) 6n; (CG-63)
C C

Uzo'lcsu- —05ue'”‘-( 2%.," >29.” (33)

The first invariant of the transformed solid strain tensor can also be obtained

by making use of the fact that according to Eq. (24) of 4.2, °mm is equal

60

to v26 .
'mmohlolo') ' '3' C5 O.”z-%k..nz (34)

The quantity 50 can also be obtained by using Eq. (2 8) of 4. 2.

 

202 (- “(Cs-3.).q‘z +(03+202)5mm°("01a.)) (35)
( l + a.) C7 202

Using the constitutive equations (1) of 7. 1 it follows that the transform of the

30 (mm) -

first invariant of the solid partial stress tensor is related to 5mmo and
5'0 as shown in Eq. (36).
Ommo("9zvs)='30|po+(202+303)§mmo (36)

The transform of the vertical partial solid stress in the half-space can be ob-

tained using the second Eq. (2) of 7. 1.

z shw’m

6 ( , , -
"o '1 “+0" m°+202 [(2'“; -5qu

(37)
0| (c 6 -QO) . 2 (c6 q+) - z 2 112
+-— 'l+ flék— Zfl n C O
(I'HI')rt C7 H’ H H‘ C7 ’20 +ll 5
The other unknowns are more difficult to obtain because their expressions
include partial derivatives of E and S , with respect to the variable r .
The derivation of explicit expressions for these other quantities is given in 8. 3.

8.3 Determining On. 0", 090 , and ur .

 

or: is defined by Eq. (38):

32 2
E a __s_
0'1 202(3razh araz —) (38)

which is one of the constitutive equations from 7. 1. Forming the Laplace, and

 

then the first order Hankel transform of this equation results in Eq. (39).

dS
6fz|'-202(qi'g dEl -+Zn— —°'dz) (39)

By Eq. (39) it can be seen that 5'1. can be expressed in terms of E0 and
§o . Equations (30) and (32) of 8. 2 provide the necessary expressions of

go and E0 that are needed for computing the derivatives which are given by

Eq. (40).

 

71-2-3— c7
‘50. «(2 k c . z - z (40)
—z'-+MC50 --? ze" -uc5 e”

Finally, 0,; can be obtained by performing inversion (41):
-I oo (IE0 d§o
On=L of-202(qu-+nz-J;-) n J|(qr)dq (41)
where l... indicates the inverse Laplace transformation is to be performed.

0: can be obtained in a similar matter.

a as
tsp-3%“? (42)

Equation (42) can be expressed in terms of known transformed quantities as
shown in Eq. (43).
ar,""lfio'z'l§o (43)
The necessary inversion process is indicated in Eq. (44).
(D
oral." [-f (qunnSo) qJ.(qr)dn (44)
o

In order to derive an expression for O" that can be computed from

EC and §o , consider Eq. (45):
§(O' _ )+(Orr‘000)+aorz'
3r " p r 82

which is the first equilibrium equation, Eq. (1) of 4. 1,expressed in cylindrical

o (45)

coordinates, using physical components. Multiplying Eq. (45) by r2 and re-

arranging terms results in Eq. (46):

a 30
5-, r2 (Crr'Pl‘r(O"-2p+0”)-r2 #’ (46)

Now if an expression can be obtained for Ono-250+ 6000 in terms of E0
and So , then Eq. (46) can be integrated with respect to r to obtain the desired

expression for O" . Such an expression follows by the definition of 0mm .

a" " 29+°00'+Omm'ozz'29 (47)
Transforming Eq. (47) yields the desired expression, Eq. (48):
°rr,-250+ 5”,: ammo-5220460 (48)

The transforms appearing on the right of Eq. (48) have been determined pre-

viously and are given by Eqs. (36), (37), and (35), respectively.

62
Returning to Eq. (46), it can be seen that the quantity 58-;(r2(0’"-p)) can

be obtained from the inversion formula (49):

 

3' Ir/L '[Ono -2§°+Ouo Jqd°(r(r)dq
(49>

-r2°f ML"[6; J]qd.(qr)dq

By making use of Eqs. (41) and (48), integrating both sides of Eq. (49) with

respect to r and interchanging the order of integration,Eq. (50) :

2 oor _| _ _
r (Omaha/of”. 0,,0-2ao+o,,o «Jolfi'lardq

r -
_ 2 ’I don.
00;]! L [_dz ]qJ|(qr)drdq

is obtained. Making use of the integrals, Eq. (51):

(50)

r
o/rJOMr) dr = 7';- (Mm)
5
off J.(qr)dr T1? «(rm-W Jam!)
results in Eq. (52):
00

on ”TH-d Lima; 250* 000°) ('7 JI(’l')d'l

(52)

coda".

     

dz. qr 2J.(qr)- Jo(qr))dr(] r>0
o

 

- :16
where 5,,0-2504-0990 is defined by Eq. (48) and '1' may be obtained
dz

by differentiating_Eq. (39) as shown in Eq. (5 3):

do?! -kc gkzc_ 2
d2 202“ (— 2‘ —+ ”'15)
+) (53)
- ’ a (c -6 -
+u2c .4." 29%;“ .m n .3— 0 .n2

Since 9 can be computed from Eq. (35), Eq. (52) is fully defined and can be
used to compute O" . Also, once a" is computed, then C“ can be

computed using omm,o" , and (522 as shownin Eq. (54):

63

000' 0mm' on " 021 (54)
All of the unknown dependent variables have now been expressed in terms

of the known transformed displacement generating functions. However, the
doubly transformed solution images that have been presented must be inverted
in order to provide a solution of the physical problem being studied. Equa-

tions (55)
Etr.s)-o/°°qavtq,s)av(m)dq (55)
and (56)
0(r,t)= L" tame» (56)
indicate symbolically the operations that will provide the physical solution.

In Eq. (55), §,(qs) denotes a typical solution image such as fit, . For the
v order Hankel inversion (55), the Laplace variable 8 is considered con-
stant, then in Eq. (56) s is allowed to vary. The actual inversion algorithm
to be used in this study is a numerical approximation of the inversion process

indicated by Eqs. (55) and (56) and is presented in Chapter IX.

IX

INVERSION OF LaPLACE-HANKEL TRANSFORMS

9. 1 General Algorithm

The Laplace-Hankel transformed solutions that were obtained in Chapter
VIII were programmed so that the transforms could be evaluated for given values
of It and s using a computer. Values of the Laplace transform of the solu-
tion are obtained at discrete points along the positive real axis in the trans-
formed plane by performing the Hankel inverse transformation with 8 held
fixed. The Hankel inversion is approximated by performing numerical inte-
grations between zeros of the integrand and accumulating the sums until succes-
sive intervals yield negligible differences. The integrand consists of the product
of the double transform with 3 fixed and the inversion kernel as shown in the
following equation:

3(r,s)-o/°°5v(q,s) “Imnavtqndn (1)

After the approximate Hankel inversion has been performed, the Laplace
transform data are used to construct a Fourier series approximation of the
time-dependent solution. The approximate solution obtained converges in the
mean to the exact solution. A truncated version of the series expansion is
used in the computer program. It is also possible to construct a Fourier series
approximation of the Laplace transform of the solution using the discrete Laplace

transform data.

64

65
There are three sources of error in the transform inversion algorithm.
In the approximate Hankel inversion, errors occur through the numerical
integration process and using finite limits of integration. In the Laplace
inversion approximation, error results because only a finite number of terms
in the Fourier series are used. Finally, error results due to round-off as in

all computer programs.

In order to investigate the accuracy of the computer program, the iterated
Laplace-Hankel transforms of several known functions were inverted using

the inversion algorithm. The results obtained agreed with the known solutions
to a reasonable degree of accuracy and indicated that the accuracy of the
algorithm is adequate. Based on the indirect accuracy checks obtained testing
known functions, it is thought that all of the algorithm errors combined are

within the accuracy of the physical definition of the problem. That is, the
errors incurred in determining mechanical constants and other physical pa-
rameters in the problem which determine the uniqueness of the mathematical

problem are consistent with those inherent in the transform inversion al-

gorithm.

More detailed information follows in 9. 2, concerning the theoretical basis

of the Iaplace inversion procedure.

9.2 Numerical Inversion of the Laplace Transform

 

The theoretical formulation of the Laplace inversion algorithm used in
this study was developed by Erdelyi [Q] . He showed that it is theoret-

ically possible to construct Fourier series approximations of functions,

66
quadradically Riemann integrable on the positive infinite internal of real num-

bers 9, utilizing only the Laplace transforms of the functions. Erdelyi ob—

-lm'

served [14, E] that the sequence of function e is linearly dense in the

vector space L2(o,oo) , which is defined as the set of quadradically integrable
functions mentioned above, if the sequence Im consists of real, positive num-

bers and satisfies Eq. (2).

1.
__I_ too
i=0 l+(li)2 (2)

He also noted that it is possible to construct an orthonormal sequence of func-

tions 4'" that is linearly dense in L2(o,oo) by applying the Gram-Schmidt

- t
orthonormalization process [gilto the functions 9 [m if the [m elements

satisfy Eq. (3).
(m’ln if min (3)
Orthonormalization of .‘lm' yields Eq. (4):
" -! t
¢ is! m 4
which defines the 45" functions.

The coefficients Cm" in Eq. (4) are defined by Eq. (5):
-l
Cmnn(2lfl)|/Z gr (!m+,l()/‘?r (I'D-IR) (5)
RD It

:0

m (him)

where n75 denotes the product of terms i through m .
n

It is, therefore, possible to construct a Fourier series representation of
any function flt) quadradically integrable on the interval zero to infinity, as
stated in Eq. (6).

:m~°2° (mnwnmn °f ( /°?(r)¢n(r)ar)¢,m
n=o use 0 (6)

 

It is shown in Appendix C that nearly all of the transformed solutions ob-
tained in Chapter VIII correspond to time-dependent solutions that have
finite steady state values not equal to zero. Such functions are not quad-
radically integrable on the infinite interval. However, if the steady state
portions of the transforms are subtracted off, the resulting transforms
correspond to quadradically integrable functions.

67

The series on the right of Eq. (6) converges to f(t) in the mean square sense
as defined in Eq. (7).

III-1i: of “nhgo (f,¢n) W1“) 2 d' 3 0 (7)
This means that H?) is approximated by the series in such a manner that the
integral of the square of the approximation error along the t axis vanishes as
the number of terms in the series becomes infinite.

The inner product terms (f,¢n) can be expressed in terms of the values

of Laplace transform of W) at sslm as shown in Eq. (8):

n
TdT 3&0 Cm" 5“,“) (8)

as n m 'I
g. m
ofnrwntrmr mgo Gama/“Tie
Substituting Eq. (8) in Eq. (6) yields an expression for f0) which can be com-
puted if sum) are known.
co n n .[ g
f 9 ~ 1 " z m 9
( ) (150‘ mm Cm" 9"m)m.o°mn o ) ( )
Series expression (9) provides an inversion formula for the Laplace trans-
formation which utilizes knowledge of the transform 6(3) at the discrete points
8",“ . Truncating the series (9) to a finite number of terms defines an
algorithm which may be used to approximate fit) if 6(3) is known.

9. 3 Improving Convergence

 

It was found in applying the algorithm to the transforms in this problem
that the partial sums of the Fourier series approximation of the solutions in
the time domain converge in an oscillatory manner at small values of time.

In order to improve the convergence, Fejer summing was performed as
defined by Eq. (10):

Fejer Surrtw'ggI Sn) / N (10)
where Sn denotes the nth partial sum of the Fourier series. This summation

procedure is known to improve the convergence of Fourier sine-cosine series L11]

68

and was formally applied to the partial sums of the generalized Fourier series

as a numerical experiment. Figure 5 shows a comparison between the con-
vergence achieved in a typical problem as a function of the number of terms in
the series by both summation methods. It can be seen that in the problem shown,
the Fej er summing procedure markedly improved the convergence. Although

no proof has been presented concerning the generality of this result, from the
numerical results obtained thus far, the Fejer summing procedure appears to

be a time saving device that provides correct results.

Another technique which yields improved convergence consists of com-
puting the mean value of the n and n—I partial sums. Results obtained using
this procedure are shown in Figure 6. This particular procedure has been
incorporated into the computer program.

9.4 Selecting 1,“ Points

 

In addition to the requirements stated in 9. 1, Erdelyi [E] suggested that
the [m sequence should be a ”base" for the Laplace transformation.” This
means that if the Laplace transformed function is approximated by some new
function and the difference between the values of the function and its approx-
imation vanish at all the "base" points, then the difference between the two
functions will vanish identically at all points indicating that the two functions
are equivalent. Further, by the uniqueness of the Laplace transformation (18)

the inverse transform of the approximating function can differ from that of. the

 

1° Any sequence of points that has a finite limit point is a base for the Laplace
transformation [13].

69

.955. wcwvoooozm ho SEE was .955

 

 

 

3:st poison .comCmQEoo oonowsoéoo .w Miami
mmem 2. gum... no «we;
n_v_n.~_._o.oosonvn~..
_a___sa________o
has". hZu_mz<E.- 0.00. a ZOFUunuuo wh(...a
923m 02.0uuUUDm no Z<u2uo
23m 45.51". "5359.3
H”:02
~o.
O O O
O O O O O O O O 0
no.
0
L:

 

53mm ’NOILDB'IJBO suns so
1w: .LN3ISNWJ. so snouivwnxosdev

69:6 .83..“ use .956

 

 

 

 

:5st sozzom 53232.8 ooccwnogoo .m menu?“
muEum z. 9.58. no mum;
n_!n_~_:o_aosenvn~..
2J___1________n.o
.EE .rzywzfiz. no.3. . zo_»Umnuuo whine
23m mufiwmuo
«23m 45.5.5 amino... .0
H952
n.
. w
No V
l.
3
a
3
J
.l
3
. n
. o m
0 0 O O O O O IN
. 1 no. w
o H
3
5
Q0.

.LUVd LNB'SNVUL JO SNOILVWIXOEddV

70

original function only by a null function which is of no consequence:
0/ Null Function (1') d‘l’ :- O , for every positive ' (11)
The practicality of placing this additional restriction on the choice of the
(m sequence becomes apparent if the Iaplace transformation is applied to
both sides of Eq. (9). Using Schwarz's inequality [lg] , it can be shown that
such a transformation will yield Eq. (12):
i(s)=§°(m°§o cm, iamlnmgocmtulmf') (12)
where the series shown converges to the transform 5(3) pointwise as defined

by Eq. (13).

fl-c-co

 

{am-i:° (f may ”onufi" dtl=0 (13)
That is, the error of approximation at each point a vanishes as the number
of terms in the series become infinite.

Conversely if the lm are restricted so as to form a base for the Laplace
transformation, then Eq. (14):
":2 EU,“ )- 750 (f, ¢n)/ q, (t) o-m'd’ =0 (14)
implies that the series approximation (12) will converge pointwise to the Laplace
transform of the unknown time-dependent solution. Therefore, since it is known
that only a finite number, of [m points can be used in numerically constructing
the Fourier series approximation (9) of f(t) , it seems logical to select the 1m

such that (14) is approximately satisfied. This can be done by requiring that

the [m be part of a sequence that is a base for the Laplace transformation.
Several 1m sequences were experimented with in developing the algorithm

for inverting the transformed solutions to the plate on half-space problem. All

of the sequences tested satisfied the requirements set forth by Erdelyi, but it

was found that numerical errors develop in the inversion process if the truncated

71

sequences that are used do not satisfy certain additional conditions.
Generally, it was found that the maximum and minimum magnitudes of the
[m points must be limited. It was also discovered that the convergence of
the [m sequence at its limit point must not be too rapid. That is, the [m
points must be spread out over the domain defined by the maximum and minimum
allowable values, rather than clustered near the limit point.
The [In sequence that was selected for use in the computer program is

defined as follows:

[0 'el

(15)

[n‘ ln-I ‘€2+€3
The parameters 6, , 62 and 63 are determined by examining the Laplace
transform image. 6' and 62 are selected to bracket the portion of the trans-
form image that maps into the transient part of the solution in the time domain.
8. defines the maximum [m value and £2 and £3 define the rate of con-
vergence of the sequence and the magnitude of the smallest [m value. Gen-
erally, 8. may be set equal to 100, 62 a,5 and G3 =,01. Figure 7
illustrates the results that were obtained applying the inversion algorithm to

the iterated Laplace-Hankel transform of a known time-dependent function using

these values of 6|: £2 and 63 .

fit)

0.5

0.4

0.3

0.2

DJ

72

 

 

O ~0*ofi § I

 

 

 

 

 

 

SERIES

‘ “" APPROXIMATION

.01 .111 .109

.05 .232 .235

SERIES .10 .313 .311

APPROXIMATION .50 .415 .413

1.00 .310 .309

5.00 .007 .006

/EXACT VALUES '0-00 000 .000

Figure 7.

ITERATED TRANSFORM =

TIME DEPENDENT FUNCTION = 6* ERF r—m

NOTE: 6.
£2
63

= I00
= .50
= .0!

1
e’n—-———
1/S+2 (S-H)

ERF DENOTES ERROR FUNCTION

 

Numerical inversion results, test function.

 

 

X

EXAMPLE PROBLEM

The following numerical example is presented to illustrate the use of the
computer program and to provide a test problem for potential program users.
Only the results of the analysis are presented here and details concern-
ing program input and output are given in the Appendix along with the program
listing.

A hypothetical pavement structure having the geometric and physical
properties indicated in Figure 8 is assumed to be subjected to the loading shown
at time zero. The foundation constants that are shown were approximated for
a saturated, well-graded sand and gravel mixture using the techniques discussed
in Appendix B-1. The stress relaxation function 0(9) that is used in this ex-
ample to describe the plate material is based on stress relaxation tests that
were performed on sand asphalt mixtures by Moavenzadeh and Soussou [_1_9_] .
Details concerning the stress relaxation data are given in Appendix B—2.

The results of the computer analysis are as follows: the reaction of the
foundation directly under the center of the load is approximately equal to 32 psi
at time zero and approaches a steady state value of 40 psi rapidly. At a dis-
tance of one load radius, or 10 in. from the origin, the foundation reaction is
16 psi at i - 0 and 19 psi at steady state. The partial fluid stress at the origin

varies from 3. 7 psi, or 11 percent of foundation reaction, at no to zero value

‘73

74

.o.~302.3m 3080.35 oEEcxm .m 0.53m

 

 

 

.0... coon unu M+~o

n~_. .1. u
a. .1. a

fin coco. .1. Nu
31.1.. a

 

 

 

 

 

 

LA...“

 

 

L.__.
I56 0'

 

rlll .<.o .8 IL

"Z/l-f

 

 

75

in steady state as shown in Figure 9. The variation of fluid pressure with depth
is shown in Figure 10. The plate deflection at the origin is shown as a function
of time in Figure 11, having an initial value of about 0. 02 in. and a steady state
value of 0. 05 in. The plate deflection at 10 in. from the origin varies from an

initial value of 0.015 in. to a steady state value of 0.03 in. as shown in Figure 12.

76

SBHONI '2 HLJBO

5000 :33 533.8.» 000.50 3:: €3.23 .3 0.3.th
on

O?

on

ON

0.

O — _ _

0.... O.N1 0.01

.0000 u 083 .002 00 0:203:00 om

 

 

 

 

.mu .mmUCPm 6.34% J<F¢<¢

 

O.‘1

Ech E 000.50 3:: 33.8%

.08.: mfimh0>

.0 0.53m

 

.mn o n u34(>
uhSrm >0<uhm

 

 

. n L P _ F

Oil

 

9N I

 

 

 

O.

o o h o 0 V
mOZOUum fl}...

lSd 'SSBULS 01m: 'IVILUVd

77

.05: 020.00., 5300 80.: 00:05 o. .00

 

 

 

00:50.0 .302 00 5:00:00 000.0% .m. 0.2%....
32003 .u
o n v n 0 _ o .
_ _ _ _ _ _ 30

0 .0.

0.0.

”9.02. among; UPS—.0 >0<u...m

1 two.

 

 

08.

1 NNO.

SBHDNI '5

 

 

.063 msmp0> szo .0 :0300C00 000.05 .3 02%....
32000.». J
o n v n N . o
. _ _ 0 _ _ o
L .0.
03. a 03.; 1 No.
m._.(...m >0<mbm
I. no.
1 3.
1 no.

 

 

 

00.

SSHONI ‘5

XI

CONC LUSION

11. 1 Summary

 

During this research a method has been developed for analyzing the
deformation response to load of a flexible pavement modeled as a linear visco-
elastic thin plate supported on a fluid filled foundation. The foundation consists
of a porous elastic solid filled with an incompressible fluid. The mathematical
model allows for an initial volume change to take place when the load is applied
in order to account for the fact that the solid particles in granular soil mate-
rial may be compressible. This allowance for initial compressibility has not
been allowed for in previous foundation settlement theories.

The model response to applied load was formulated as an initial-boundary
value problem. The technique used to solve the problem consisted of reducing
the partial differential equations of the problem to ordinary differential equa-
tions using transform methods. The transformed problem was solved yield-
ing iterated Laplace-Hankel transformed images of the desired solutions. A
numerical technique was developed to invert the transforms and obtain the
desired physical solutions, the stresses and displacements in the model. The
results of the analysis have been recorded in a computer program. Using the

program, it is possible to compute plate deflections, foundation to plate reactions,

78

79

fluid pressures, and eight other stress or displacement quantities as functions
of time. The program is designed to be used by others, and instructions con-
cerning its use are given in Appendix A. It is noted that the portion of the pro-
gram that was developed for inverting Laplace-Hankel transforms is applicable
to other problems involving the inversion of either iterated, or single trans-
forms of the types mentioned here.
11. 2 Further Study

The most difficult problem encountered when one attempts to estimate
stresses and displacements in an actual pavement structure utilizing the devel-
oped computer program is that of determining the values of the material con-
stants. Stress relaxation data are required to describe the plate material, and
uniaxial stress relaxation tests at constant temperature are required to deter-
mine the needed data. The data currently available in the literature need to
be extended. There are also five poro-elastic constants which are used in the
analysis. Suggested tests for determining these constants are presented in Ap-
pendix B-l. Several tests were made to determine the poro-elastic constants
for a typical base course material for use in the sample problem analysis given
in Chapter X. The tests were performed using a triaxial soil testing device.
From the limited experience gained in performing these tests it appears that
although the tests are difficult to perform, such testing is feasible. The poro-
elastic properties of base course and subbase materials need to be investigated
extensively.

After the poro-elastic and viscoelastic material properties of highway
pavement materials have been investigated adequately ,. it will be feasible to

check the accuracy of the flexible pavement model. Such verification should

80

include plate load testing of real pavement sections having known material
properties. If the model proves accurate for simple pavement structures
consisting of a single layer of bituminous material resting on a single founda-
tion material, it would then be useful to extend the model to account for layers
of different porous media occurring in the foundation. Another phenomenon
that will require investigation is the effect of partial saturation on the founda-
tion. If isothermal deformation is assumed, it may be possible to allow for

this in the mathematical model by treating the pore fluid as compressible [l]

REFERENCES

REFERENCES

1. Biot M. A. , "General Theory of Three—Dimensional Consolidation, "
Journal of Applied Physics, Vol. 12, 1941, pp. 155-164.

 

2. Terzaghi, K. , Erdbaumechanik auf Bodenphysikalischer Grundlage,
Leipzig,F. Deuticke, 1925.

3. Paris, G. , "Flow of Fluids through Porous Deformable Solids, "
Applied Mechanics Survey, Washington, Spartan Books, 1966.

 

4. Green, A. E. , and Naghdi, P. M. , "A Dynamical Theory of Inter-
acting Continua, " International Journal of Engineering Science, Vol. 3,
1965, pp. 231—241.

 

5. Tabaddor, Farhad, "Constitutive Equations and the Solution of Some
Problems of Interacting Continuous Media, " Ph. D. Thesis, Michigan State
University, 1968.

6. Green, A. E., and Steel, T. R., "Constitutive Equations for Inter-—
acting Continua, ” International Journal of Engineering Science, Vol. 4, 1966,
pp. 483-500.

 

7. Biot, M. A. , and Willis, D. G., "The Elastic Coefficients of the
Theory of Consolidation, " Journal of Applied Mechanics, December 1967,
pp. 594-601.

 

8. McNamee, J. and Gibson, R. E. , "Displacement Functions and
Linear Transforms Applied to Diffusion through POrous Elastic Media, "
Quarterly Journal Mechanics and Applied Mathematics, Vol. XIII. Pt- 1
1960, pp. 98-111.

 

9. Westmann, R. A. , "Viscoelastic Layered System Subjected to Moving
Loads, " Journal of Engineering Mechanics Division, Proceedings of ASCE,
June, 1967, pp. 201-218.

 

10. Hoskin, B. C. , and Lee, E. H. , "Flexible Surface on Viscoelastic
Sub-grades, " Journal of Engineering Mechanics Division, Proceedings of
ASCE, October, 1959, pp. 11-30.

 

11. Timoshenko, S. , Theory of Elasticity, McGraw-Hill, New York,
1934 , p. 310.

 

81

82

12. Sncddnn, I. N., Fourier Transforms, pp. 469—470 and 454—455,
McGraw-Hill, 1951.

 

l3. Erdelyi, A. , "Note on an Inversion Formula for the Laplace 'I‘rans»
formation, " The Journal of the London Mathematical Society, Vol. XVIII,
1943, pp. 72-77.

14. Szasz, O. , "Uber die Approximation stetiger Funktionen durch
lineare Aggregate von Potenzen, " Math. Ann. , 77(1916), pp. 482-496.

 

15. Shohat, "Laguerre polynomials and the Laplace transforms, " Duke
Mathematics Journal, 6 (1940), pp. 615-626.

 

16. Indritz, J. , Methods in Analysis, Macmillan Company, New York,
1963 , pp. 15-16 and p. 242.

 

17. Lanczos, C., Linear Differential Operations, van Nostrand, 1961,
pp. 71-740

 

18. Churchill, R. V., Operational Mathematics, McGraw-Hill, New
York, 1958 , p. 183.

 

19. Moavenzadeh, F. , and Soussou, J. , "Viscoelastic Constitutive
Equations for Sand-Asphalt Mixtures, " Highway Research Record, No. 256,
1968, pp. 36-52.

 

20. Pagan, C. , "Rheological Response of Bituminous Concrete, " High-
way Research Record No. 67, 1965, pp. 1-26.

 

21. Secor, K. , Monismith, C. , "Viscoelastic Response of Asphalt
Paving Slabs Under Creep Loading, " Highway Research Record No. 67 ,
1965, pp. 84-970

 

22. Huang, Y. , "Deformation and Volume Change Characteristics of a
Sand-Asphalt Mixture Under Constant Direct and Triaxial Compressive
Stresses, " Virginia Highway Research Council, Reprint No. 65, April 1968.

23. Schapery, R. A. , "Approximate Methods of Transform Inversion
of Viscoelastic Stress Analysis, " Proceedings, 4th U. S. National Congress
of Applied Mechanics, 1962, pp. 1075-1085.

 

24. Korn, G. , and Korn, T. , Mathematical Handbook for Scientists
and Engineers, p. 228, McGraw-Hill, 1968.

 

25. Pickett, G., Raville, M., Janes, W. , and McCormick, F., "De-
flections, Moments, and Reactive Pressures for Concrete Pavements,"
Kansas State College Bulletin, No. 65, 1951.

APPENDIX

APPENDIX A

COMPUTER PROGRAM LISTING AND USER GUIDE

What the Program Does

 

The program determines the magnitude of any stress or displacement com-
ponent at any appropriate position in the plate on half-space flexible pavement
mooel as a funCtion of time. The user must specify which unknown Stress or
displacement function is required, loading conditions, the magnitudes of mate-
rial constants, and the position where the solution is to be determined.

Using the input information specified, and function sub—routines, the pro-
gram constructs discrete values of the Laplace-Hankel transformed image of
the desired solution. These data are integrated numerically for fixed values
of the Laplace variable to yield discrete values of the Laplace transformed
image of the solution. (This numerical integration process approximates the
exact Hankel transform inversion operator. ) The program then uses the dis-
crete data to numerically invert the Laplace transformed solution to obtain
the desired physical solution. The Laplace inversion process that is used
consists of constructing a Fourier series approximation of the time -dependent
solution using the discrete Laplace transform data. The Fourier series ap-
proximates the exact solution in the mean square sense, and its Laplace trans-

form approximates the Laplace transform of the solution pointwise.

83

84

Execution, or process time, on this program varies from two to ten min-
utes for one set of data.

The Program ALGORITHM

 

The algorithm begins in the main program where SET UP is called. In
SET UP the problem data is read in and then printed out. The zeroes of the
appropriate Hankel kernel are also computed in SET UP and stored in the arrav
ATJ ( ). Control then returns to the main program and the steady state and
initial values are obtained and stored in STEADY and F(1). To do this the main
program calls HANKEL setting GCODE equal to 2 and 3 and s equal to 1. Util-
izing the ATJ ( ), HANKEL calls SMPSM repeatedly where numerical inte-
gration is performed on the inverse Laplace transform of the desired solutions
between successive ATJ ( ) points. Within SM PSN the function G is called
which in turn calls one of the G 1 through G 11 functions in order to evaluate
the integral as required for numerical integration. The integral results be—
tween successive ATJ ( ) are accumulated in HANKEL until additional inte-
grations contribute less then DEL x 100% of the accumulated results.

The main program then determines the transient part of the solution. The
L ( ) sequence and C ( , ) array are computed in the main program and
are used to construct a set of orthonormal functions using the Gram-Schmidt
process. The Laplace transform of the desired solution is then determined
at each of the L ( ) points and is stored in GINTP ( ). This is done by
setting GCODE equal to 1 and 8 equal to successive values of L ( ) and calling
function HANKEL where SMPSN is called as was done in determining STEADY

and F(1).

85
The main program utilizes C ( , ), L ( ), and GINTP ( ) to con-

struct a truncated Fourier series approximation of the transient part of the
solution for each specified value of time, T ( ). The transient solution is
stored in F(2) through F (KK) and then combined with STEADY to yield a NN
term series approximation of the problem solution, Q ( ) which is then output.
Mean sum approximations, wnich are defined as the sum of the NN and NN-l
approximations of Q ( ) divided by two, are also computed and are stored in
QMEAN ( ). Solution approximation results are output consisting of F ( ),
Q ( ) and QMEAN ( ) for each NN series approximation. This process is
repeated NN equal I through NNN and successive series approximations are
compared. The process is terminated before N NN is reached if a succeeding
approximation differs from the previous by less than SUMDEL x 100%- The final
QMEAN ( ) approximation is output as the solution along with the times

T ( ).

Card Input Description

Order of Precedence as given:

Variable Names Format Optional
MG, OPT1, OP'I‘2, OPT3 412 No
K, (T (I), I=2, KK) 12,8X,7D10.5/(8D10.5) Yes
DEL, BJDEL, IMAX, NJO, SUMDEL F10.5, F16.10, 212, F10.5 Yes
L1,ACCPT, FOCUS, NNN 3D6.3,12 Yes
Z, RR, H, BE, Q 5F10.5 No
FF, R, BULK, GAMA, DARCY, A2 2F10.5, 4E10.4 No
R0, NB E10.4, 12 No
B(I), 1= 1, NB 8E10.4/ No

1313(1), 1= 1, NB 8E10.4/ No

Variable Description

 

86

OPT1, OPT2, OPT3 determine which optional input cards are being utilized.

MG

Setting OPT1, OPT2, and OPT3 equal to zero indicates

that no optional data cards are being used. Setting OP'I‘l

equal to one indicates that the time array is to be specified.

Setting OPTZ equal to one indicates that DE L, BJDE L, IN] AN,

NJO and SUMDEL are specified. Setting OPT3 equal to (mu

indicates L1, FOCUS, ACCPT, and NNN are to be speciliud.

determines which stress or displacement component is to

be computed. When it has a value

the foundation reaction stress at the plate boundary
is to be computed.

the plate deflection is to be computed at some point.
implies the vertical displacement component at some
point in the foundation is to be computed.

implies the vertical stress component is to be
computed.

implies the first invariant of the solid strain tensor
is to be computed.

implies the partial fluid stress is to be computed.

implies the first invariant of the solid partial stress
tensor is to be computed.
implies the solid radial displacement component is

to be computed.

87

=9, implies the shear stress, Oz, is to be computed.
=10, implies the radial stress component of the solid

partial stress tensor is to be calculated.

11, implies the tangential component, 0” , of the

solid partial stress tensor is to be calculated.

KK indicates the number of time values at which the solution
is desired.

T (I) the time array, having KK elements, the first of which is
automatically specified as zero in the program. Only KK-l
values may be read in as input.

DEL is a decimal tolerance factor. DEL COHtPOlS the accuracy
of the numerical integration performed in sub-routine
function HANKE L, but not the accuracy of the numerical
Laplace transform inversion that is performed in the main
program.

BJDEL is a decimal tolerance factor. BJDEL controls the ac-
curacy of the Bessel function values that are computed in
sub-routine function BJ ( , ).

IMAX limits the number of points used in performing numerical

integration over a given interval using sub-routine SM PSN.

NJO limits the maximum number partitions that may be used in
sub-routine function HANKEL in performing the numerical
Hankel transform inversion process.

SUMDEL is tolerance factor that controls number of terms used in

series approximations of solution.

ACCPT

FOCUS

NNN

BE

FF

BULK

GAMA

88

is largest element L(1) in L ( ) array.

is equal to .5 times the limit point of L ( ) array.

is a constant that controls the convergence of the L ( )
array. It is greater than . 1 and less than 1.

limits the number orthonormal vectors that may be con-
structed for use in the Fourier series approximation of
Laplace transforms in the main program. NNN is the
maximum number of terms that will be allowed in the
truncated Fourier series.

is the vertical depth to a given point in the model. Z is
measured positive downward and is equal to zero at the
plate - half-space interface.

is the horizontal radial distance from the axis of the applied
load to a point in the model.

is the plate thickness. This value is used in computing the
bending stiffness of the plate.

is the radius of the loaded circular area.

is the applied load pressure.

is the porosity of the foundation material.

is the pore compressibility constant.

is the apparent bulk modulus of the porous media in the
steady state.

is the fluid density of fluid component of the interacting

mixture.

DARCY

RO

B (1)

BB (I)

Sample Input

 

Problem:

89

is the Darcy coefficient of permeability of the porous
medium.

is the shear modulus of the solid material.

is the initial value of the stress relaxation function of the
plate material.

is the number of exponential terms that are to be used

to describe the stress relaxation function of the plate
material.

is an array that contains the exponents of the exponential
terms used in relaxation function.

is an array that contains the coefficients of the exponential

terms used in the relaxation function.

Utilizing the foundation and plate material properties given in this report,

determine the plate deflection directly under the loaded area at the origin of the

model.

The loading pressure is assumed to be 40 psi and the radius of the loaded

area, 10 in. The plate is assumed to be 4-1/2 in. thick. The solution to this

problem is given in Chapter X .

The required input is as follows:

MG

OPT1

OPTZ

OPT3

90

RR = 0 in
H = 4.5 in.
BE = 10. in.
Q = 40. psi
FF = .125 '
R = .9
BULK = 3000.
GAMA = .0361
DARCY = . 196
A2 = 10000.
R0 = 156029.
NB = 11

B(I) I=2, NB = 5000. , 500. , 50. , 5. , .5, .05, .005, .0005, . 00005, .000005
BB(I) I=2, NG= 6650, 23200, 41900, 33000, 30100, 12600, 4300, 1440, 836, 403
Note: The option to use no optional cards has been chosen, therefore, certain var—
iables will be automatically specified in the program as follows:

KK = 20

T(I), I= 1, KK O, .01, .05, .1, .25, .5, 1., 1.5, 2., 2.5, 3.0, 3.5, 4.,

4.5, 5., 7., 10., 15., 20., 30.

DEL = 0. 001
BJDEL = . 001
SUMDEL = . 05
IMAX = 8
NJO = 60
NNN = 14

L(1) 1.D2

91

FOCUS . 5

.01

ACCP’I‘

The required input cards appear as shown in Figure 1 .

A complete program listing follows these instructions. Comment cards
have been inserted in the listing to explain various parts of the program.

Optional Input Data

 

Certain input data are optional in that- the user need not specify these items
unless he desires. The input is set up in this way to minimize the required in-
put but to also insure maximum program flexibility

Special Cases

 

0", RR I O:
In this case it is necessary to submit an extra set of input cards with MG

set equal to 12. The reason for this is that in this case it is necessary to per—
form two separate integrations in computing the Laplace transform of on. .
One integration is performed on G12, and the other integration is performed on
G10.

If 1 =RR= Othen only one set of data is required with MG set equal to 10.

O”,RRIIO:

The program will not solve for 0” directly. However, 0“ may be
obtained by solving for 0mm , 022 and O" separately using the program,

and then combining the results according to the following formula.
000 ' 0mm ‘ Orr " c’zz
If RR8 0 .z , then G11 may be obtained directly by setting MG equal

to 11.

 

 

92

 

  
  

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MAIN PROGRAM FOR

FLEXIBLE PAVEMENT MODE L

93

S94

 

 

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99

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FUNCTION HANKE L

105

106

 

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SUBROUTINE SM PSN

107

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IIxI zIII :cIquIwzcu II cum: muzIIzoc uuxIc II orIcIuaII

IIIIIIIIII .IIIIIIII IIIIII IIII IIIIIII II III. Im:x I IxI
I.IIII.I-..IIII.IIII.I.II.I.IIm.Iwc.m.I chquucI uIaccc

.IIOIICI zusz IzwaIIIm
IIIIIIIII IIIIuo IxI IoaI IIIIIII we cIaoxm I zaanu II I

III .IIIIIII II urIIIua meI Ic chIII> zoImIIucI uImsoo I II

vaIIII.IzI¢IIIvIIIII.Iuo.¢.I Izmnam uzIIzcccaw

.IIIIIIIII IIIIIIIIII IxI IIIIzIaIuI IIIII

IIII cquI wgaIcquI uaI quuc II uaaIIIu .Iuo zIzI mmuI
II IIIIIIzI III II IIIIII IIImmwuusm zuIIIII uIzIaIIAIc
IIIII czIIII: II>¢II2I JIII owacoucuu II IIDI wuromaaIa

ocquI

 

 

zmaam

cI IIIu zI cuImII mIIcecuIII¢ chquII no vuaIz uzIzIIIzgu
IzuyuIIIn IIIIIIII IIIIIou III; III: IIcIcII czIIIII

.a czI I

zuuqum x «cu IIII quoIIcu III): aIaooacaav chqu:I II

oquIauc maIaccaIanm 2eIIu22I czI muzIIacoc:v

u>ch wuccu acacu III

wanaua

mIIIm xIaI zI II: III IIIIIIII IIIIIIII IIawI

n zIJI vaI IIII nIcII

OIIuc ~I¢II

III IIcuI

aoccu c2 OIcuI
wcuxn woou «ovum IzIIIamucIcuI
I IzIIIIacI zI III: IIIIIIIII II IIIIII IzIII:IIII z
IIacuIzI no szI> IIrIu IrIII:mu¢I v

2cIII=II>u
IIzII II aeIII II:I IIIIIIzI Io uzII> IzIIIzmuco III
IzII> IIIIIIII Io mchIIIIIIouII Ic aways: aaaIxIaIxIII
uuchuIcI Io IIIcaqu cquzoucI Iuc
IIaII chIIccuIyI IIIIII a
IIIII chIIIeuIzI IIIOII I
IIII IIIIII IIIIIIIIII chIIzII cum: I: IaIz. I

IIUIIIIIII I: zoIIIIaumuc

IIII.I.I.III.IIII.III.I.I.IIIIIII IIII

IcIm:

uc2¢¢ cquauvwan I1» ¢u>c ICIIszu zu>~c u!» muhczcuwz—

umcac:u

zmIar uzIIIccczm

 

 

 

Ill. IIIIL

1"..- ."

LN" -

CI

\

10f)

 

can

28:er

IIIszaIIz

OIIuI

cm CI cc

IIIII

.I I «II: I new:

bat—IZOU

tn

om

«A

 

 

 

IIIzmIa IIIIIIIczI xIII uxI IIIIIIcI

Ia:¢IvIv
IuIzzII.nIxImIIIIxasmeazr
Imm.xquIxaawaaav In
uquIxxIxx
ImIIIIIII In cc
quzI\IaIu2II
IIIIIszIwIIx
xImcquu
Iwm.IIm¢IIcha:v
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nIIIv

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Im4<>¢wbzn Icon-IIIIQIN c»2~ IRIII ucI>Ic

Io \ «I I IawmIqu I awn-«Ia I I raam I r
In \.~ I «cc ImmeIo I xaav
IIIJIII

JII~\¢III rn

III¢1¢~n upanaou

zanIuc
nIIuI Iu
n~II~II~IIIxIIIIII In

Icahn:
~ICU~ «N

U

u

 

 

 

 

FUNCTIONS G THROUGH G 12

.110

111

 

unzwpzou
Athavre I mpqmu
av.»c.~u I mpI~c
coo c».co a.n.ou.uccuc. u—
uzznpzcu
rczpuc
amqp<unc I a
u:2~»zou
.vIIvao ImIIwc
crn c» cc AIp.oquccuc. nu
uzz—hzcu
zcrpwc
AmI»I.~c I c
zczpuc
athIv—c I c
cIIncco—Ioc_F.cco~.oco.ooo.oos.ooo.oom.occ.ocn.ooono—v c» cw
Izznpzcu
u:z—»2cu
“I.MIIn.I.~.IIII»I I ~u I o I .nIIquI II I «I I.w . I av
\ “Isaac I ma I o I ~II.~. I mwxc
.III»~.I..IIpsovIquIII.~v\ “aruIncququIIIN I an.»I.—¢.I mu
3» w<- v I »4 IIwu \ as I no
so \ .amIva—wucuu I n:
«m c» ca ..o.ou.pq.u~
co 0» ea .nIou.uccuc.u~
u=znhzou
.an I . II I IN . I IN. x».
an.InIII~IrII~IIINIn~IIpn I ~II»<I~II.~ I nIIhI I ~IIa~<I.mu. I su
awII—hII»-.I p~yx n>~Ibcoomw I «I I xCI on
one» I cc I ut

u:z—»zeu

cm.

can

can

can

ccm

co“

co

.n

or

en

 

.c I cue»
522—»2cu I.

anIanI cowNMIvIIIIII.~I.~II.~Iccu~nI. I IIIIIIoIIcIu. .
x I wcuwc I oI ucInIIIIIxc
u:2_»:ou I.

I— c» cc

. u..~IIch~nIy I III.«. \ .nII um I a I «I. I II

nuIanIIMIUI~<II~VxIwIIumI0I~¢. I mvxc

w" c» co Aco>e.»¢. u—

.~»~vuxu I ~p~uuu

n~>¢utxu I ~H¢wa

N I hue I Ip~

~ I wII I n»-

u x m I IIFN

.umwpcvuccm I »~

ex» I «II». I gnu».

.—.1cx.sc I ucusc

pcIuc I we:

a—Ificxufic I ccufic

“c.3cxusc I IIcwc

pIIII I we:
u22~hacu :—

Icwm

II». .uon.n.u».cx

“Cqu~uIIu ano—Ic—UIIbcxnI In» bdaICB «cr

 

an—chwv \ nuucc I mo» I var

._.II I can» I are»

 

 

0‘ or on a OIoIIcIm. a»
mucoueIupIsc.mpIoovaInIIIIIIoIIIIuc I
.nu.~»~cyu.~»IIxu.~I-.um—wo.¢¢—wc.cccwc.nmxI \2350\ n
uccuc \ucoas ~
IIIN
ccu~n¢.~III¢IoIIJIO¢IDIIxapn.o.«aceIxozoqursIuoIn.xIInud -

InchIsuIcuIb~IcIuIQ¢I~¢I«IIIquIaouuuoIaouvc \I<I¢I\ zcaiou

nnIhduu ao—hurak

 

 

 

AVI>1VCC

A’ID.§\I.I

:.I~>?Cb CFC—

I II»..IL

' (sCKok

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«VIIQIPG I C

\IIIth II?» RI

Ixus'

1L12

 

 

 

nocuc \wcozx N

az.~

cogs»I.~c.aa.o:.4.oa.42.I44¢.o.IaIc.xozou.me.ma.~.xu.n~I a

.uL».II.Iu.»~.r.I.n..~¢.II.I.Iu.no..ma..c—.a

\IIIII\ zczaou

nthcvao zo~ru23u

czu

zaspuc
an.»cv~—cIo
uzz—rzcu
nmIpcuIc I mrccu
nvI><vsu I abuse
.m.Ȣ.oc I mpIoc
.v.»Iuno I mIImc

nvI>¢v~¢ I mhINc

cmn~ o» co «IMIOquccuo. a.

uvzupzcu

zazpuc

am.pI...a I c
uzzupzou
.m.»<.o—eIw»Ic—c
nmIpcvce I mrIcc
nvI>¢~su I mrIsc
Aqucuoo I thoc
nthcvna I mhcno

nth<~Nu I mbqwc

In." o» ca ..n.ou.uc{uu. I.

wrapprcu

zappuc

an.»¢.c—e I e
u:z~»2oo
,«.»IvIu I IIIIc

athCVsc I vhcsc

cow—

cow—

emu“

coa—

emc—

 

 

AmIpcvwe I mwcwe
coo. c» ca ..n.ou.uccuc. I—
w:z—»aou
zappuc
an.»a.oc I c
uzzuhzcu
awIpcuwa I mhIwe
coo c» cu n.n.ow.uccue. u—
u:2_»2cu
zcrpuc
AmIpcvcw I e
max—pron
.mI><u~¢ I upcmc
cos c» co “.n.ou.uccuc. I—
uazupzcu
zczpuz
.thcvsc I c
wszppzou
amIrcuoa I mhcoe
athcvma I mpInc
nmIp¢u~¢ I mpre
cos c» cc "InIaquccuev up
u:2~»zou
2c:»wc
anscvco I e
u:z_»2au
athcuno I mbcnc
am.p«-e I nIINu
one c» co .IMIoqucoua. a»
wazuwzcu
scape:

athauna I c

coca

(no

cca

on:

cc:

cns

cos

cno

coo

uzz~pzou com

 

 

 

.r..¢.mc

.IC- h kirk

\ \

I;IC.QIILI\:IVIICCIIIIIEIVIIIQIIIQRV

113

 

 

IC- irhnlfluflflc I—I FdICCh

so \ noninthcuwcv I we

mrzuwvou

wooucI.00nIco~Ioo~. c» cc

In I I. I «I I.~c \ .nIInII.«. I u I mmxc I IcoU
u32~hrcu

z¢:»wu

n.0IUInIIquIIIoII.I. \ .ccuNnII~II.~v I umeu I a I no

c I h. I o I wan»

unzuwzou

zczrwa

AIMIInIUIwInII.I. \ .nIImIIINV I chwa I c I we
uqu OI». Ipzuczuauozu ux_»

cazupzou

zczpuc

IanIImI.ImeIIII.\nnII~IIIIqucIuaIOIxccu

“IocuInIINIVIIInIIII. \ accu~pII~III~u I uquc I o I we
uqu OI». .hzuczunuc uaup
war—>209

wooucIACMIoon—v c» ow

on 0» cc aoIkthqc u—

m»¢o—cIv»IscImw¢ocIv»ancIw»IIcIm»¢~c I
IIUIIINIIU.NIIIIUIIII~Iuo—IIIIIIIIIIIoIIIImII \z:ue\ n

matuc \ucoxx w

IzIw

IIIKIIIquchuzII.o¢IJIrIJQIIo.IIIchozouImcqucINIchnmc a
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.m.»..~e Ic~puzau

(0'

cc"

cm

cm

cm

0—

 

 

 

a

 

I I II I cccfic I IIIIIcIIIII. I I I III. a .I
o I ua~p ecu ~u

cor—IICc

zczuuc

¢¢0fi¢IbdoncoU I —c

uqu pauczuawozu ua~»

u:2~»7cu
zazruc

«Iowa I II I “c I .c

uqu pzuozuauc an,» m. IDJuc

Icahn: ._.uz.ua. I.
kyuzzuac aux»: 2» mm: «on «c no u24I> m~ uzch

Duo I 8—6 I —¢
Ic.u I I..\ «"9 I run I I.c
a“ cue I so “In I vax ucufim Inc I o I an I c—c

IIIFIInmCIIm I "I. I c II_c

u:z~»7cu
cocoa-«cenIocwIconv c» cc
n \ .nIIva I u I mmxcu I Iccu
u:z—»:cc
zc;Iuc
ICI—c
c I II I o I wan»
uzz~p2cc
uzr—pzou
uzz—pzcu
ucoucIncn.onI&.v c» cc
ea c» cc ac.»u.»I. a.

IIIoIIIIIIIc.v»IIcIIIIncIIIIIc.n»I~c

«cm

cow

cc—

cm

on
cw

c—

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a I ua~u Ion me

w>-~aom I II I hzucxutuoz— wI—h

nae—I

zc_>uzau «ma»: 1— mm: sou no

NI ~><wa I . mhconhc I

.114

NINIxu I

IIIIIu I .

uocuc
In I.~. I I I IIIqu I
. I I II I ,IIIII. \ .~IIII._. I .nIIuII.~

u>-~muc I h: I hzwc2UJuc wauh

o I >¢ I o I usuh

.InIIIIyImIIIIIIV I

o I II I IzuozIIuoI— uI~I

 

m \ a IN\~I a
Ina I u I rm I crnc

waznhzcu con

2:3»uc

can I p. I uccu I nc

uazuhzou cow
zcapuc

uzzupzcu oo—

o.u.xw.II x“. IIIIOI coo
canwmIpccncInc
zcapuc .I.uz.ua. II

are I saw I Inc I nc
nmIInvIUIxc u I one
anuIIIV I Inc

wbcac I nUIhuu I Inc

wrzuhzou ecu

InoCnIocono—u c» cc

. r Inga. I Ixou

unznpzou cm

zczhuc

..ocuInII~I.ImIIIIII. \ AccuunIIIIIIII I ucIuI I a I nc

unzupch Cr

zcabwc

“LIIIII.~. I IcIuo I o I ”I .

 

 

 

war—Iron cu
Icapuc
..nIIII.IIIIII.I.I.»IIIII.I.IuqucIoI-Iou
.IcguunIIII.\.I I «I I.~.I..»III.I ucIuI I o. I nc

CI». I htuczuswo wan»

usx—hzcu cu
uccuoIacnIoonuu Ch cc

on c» cu athqucv L—

mIIOIcIanIcIanocvaIncIanIcInhINc I
Inu.IIIIII.NIIIquIIINInc—IIIIIIIIIIIoIIIIIII \2:Iu\ I

monuc \uooxx ~

It.“

CIINIII~III¢IuIIxIC¢I3IIx42quIIaIcIrozcquuIIonnIchnNI —

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IIIIIcno zc~qu=I

czu
II:IUI
cacac I II I cpwc I me
am I II I «I III. \ nocu~nI I «I III. I u I II I CINc
o I waub can mm
uazupacu con
:c:»ut

IcochII I Iccu I «c

u>~>~mcIIpI Ipzuczunwoz— wan»
Izznpzou co~
za:»u¢

IxcchIIIch «c

w>~»_mcII»I .Izuczwuuc uI~I

Icahuc aaquch. 5—

 

 

 

OO'Iu‘.oA'F(O'»I'.1L

\

\

I'IIL \II

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unzIIzcu oo—

uocuu..pcn.oo~.oo~I oI ow

I I NIIIIu I u I IIIII I Iccu

UDzIIzcu cm

zcaIuI

Io I no
0 I II I o I uaIw
o I II I IzuozIIuoII uaII
CIII I Izuczucwo wauI

cm DI ca IoIIcIII. an

mIIcIcImIIIoImIIooIvIIoeImIIIuImIIne . I
InuIIIIIIuIIIIIIIIIIINIIIIIIIIIIIIIIIOIIIIIII Izauux n

uocuo \uoons w

«III

ocwnnIIIIIIcIuIIzIocIaaIsaacIoIIIIcIIozouImchucINIIIIINI I

ImoIIIIIIuIIIIcIuInIIwII.IIIIIIIIOIIIII.oIII \IIIIIx IcIIOu

IIIIIIII chqunu

czu

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crown I II I GIIu I we

I \ IuI I IIIIII. . I

IIOIUIIII.IIIIII.o¢I~III~III~IIIm I NIIIIu I u I III I cIIc
o I uauI Ion Ia .

wax—Izcu

zzquI

cacao I II I Icou I II

uIIIIIOI I II I Izuozquoz~ UIII

u:z—I2cu
tarhuc

«coscIIIIIcIIc

can

onw

 

 

 

.oIIquIIIuXmI z—IIIIICI IOI
III I III I II
zo~quau quIo II um: IcI Ia
.IIII I «III I IIIo. ImII.~ I III
NINIxu I nu I «IIII I IIIo
~IIIIUI~ I ImIImcIIIIIIIIINIIIInvxxc I II. I «3.0
~IIIxu I .III I .III
IIIII—I \ «Iqu I II I «I I I
.nu I II I II. I ..III~I.~. \ III. I "III
._II._. I IIInu I .wIIIII.~InI. I III
u’_I~m9I I II I Izwczquo uaII
I=2IIzou III
uocquIoonIocquoI. cI cu
m \ ~IIIxu I u I I
.NIII I ~I I IIIIIIINII .IIINIIINII ~III.I IIIII. I rang
unzIIzcu cm
II:IUI
.0 I II
c I II I o I UIII
c I II I IzuczquOII uIII
cIII I IzuczuIIc IIII

on OI ca acIIuIIII II

IIIoIIImIIIcIIIIIeIIIIIIIIIIIIIIIIwc I
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uooue \uc01\ «

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OIINIIIIIIIIIIIIIIIIIaaInga.IoIIIIIIIozouIIIIIuIINIIIInuI I

IIcIIIIIIuIIIIcIoInIIuIIIIIIIIIIao—IIIIIo..I \IIIII\ zoaaou

IIIIIIII zoIquaa

cau

2c:Iu¢

 

 

 

 

bttIIICL CC-

IOCUGII 0C! ICCQICOI I CI CC

 

IIIN

\

NIIIIB

rt.I

 

1113

 

 

c o r. . panels-5:1. haur
co». . ptuczunuo man»

on a» co ac.»o.»¢~ nu

upcodoqn~¢sc.opcoo.rp¢no.npccc.n»¢~c n
.nu.s»~.-u.~»..uu.p.»~.ucvao..-—so.ccono.nnxu \zagox n

accuo \ucoxx ~

az.~

ccunr..~c.cc.oa.x.oc.aa.x4:o.o.¢:¢¢.xozou.mcg.un.~.xz.n~¢ —

.vu».su.¢u.»~.¢.u.n..~.....¢.uu..o..oo..o..¢ xx¢c¢;\ :oaaou

.m.»..»o zappuzau

czu
zeapuc
econ; . p. . open]. cc
“moA—coouuv \ aocu~n¢ o n... o Nrccxu a u o a: a crow
o I man» con 09
wazupzau
2::huc
x8990. o¢
u>~>_no; 0 >4 . pzuczuuuoz_ wrap
max—pzcu
zuapuc
cccac.»«.oonuoc
acapuc .o.uz.ux. nu
.o—.o—u.oruxn. xuupcacou
.u.o.~.\~¢o.mcancu o ¢ou~n do

rc—ruzau tux»: zu am: can 00

-..-u . uptmc . p. . .oo
a~¢o.~.\ npqno . a~¢¢.~.n¢. a zoo

u>~p~mug - u. . ~auczu¢uo ma.»
u:z~»2cu

macaw-accu.oc~.oouu c» cc

can

con

ncs

co—

 

 

 

Icahn:

.c . cc
0 a pa . ptucruguoz~ ma.»
cu». . pzurzu;uo as.»
o a pa ‘ o a man»

on Ch :0 aoopeobcw up

upco—o-u>-s¢.vpcoagv»¢n¢.mpccn.n»¢~¢ -
.nu.~»~;rw.~pc‘uu.»<»~.uc_wu.¢¢—wc.¢cown.wmxa \2356\ p

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ccuuna.~c-ctgo).1.ctgaagsasw.o.¢ac¢cxozou-n8tguc.~qxlqnwd —

.th-sc.ou.h-c.u.nc-Ncn'cgcguu-ao—uconacnvc \a¢¢¢c\ roaaou

an.»c.cc 2c~buzau

can
zczpuc
3:05: o w. o cpro a re
a \ aȢ;uu o u o 2:. - cpac
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uzr—pzoe
uccuo.aoOn.oc~.oc.. c» cc
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u>~pumoc - vc . pluczwcuoru ua—u
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rczpuc
crossoraore-ae
zcapuc .r.uz.ca. a.
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zo—pulau cuxpo In was 3:; he

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can

can

 

 

 

117

 

 

 

.Ic».su.oo.»~.c.u.nI.~I..I.I.Iu..o¢.on..o..o \IIIII\ :caaou

.a.»I.II zo~puxas

cxu
Issua-
ccufio I b. I cram I no
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o I uauu Io. Id
wig—Izou eon
acapuc
cc—fio I >1 I xcou I ma
u>~p~noI I II I pruozquoz~ uxup
unzuhzcu cow
Icarus
cc—fioIhIIceIoa
Icarus aququtu up
acququIcoxnI zuquaaou nos
cow I Inc I oe
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thIxu I «on I «no
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untapzou ecu
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a ~I». I IIIII I «IInan I .nIInII.~. I NIpI I max: I I xcou
untapzou on
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I0 I no
c I »< I o I Haw»
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CIuc I bzuczunwo wan»

 

 

on a» co .o.u¢.»I. I.

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Inc».suIIo.»~IcIu.nI.uI..III.Iu..o..oI..o—.I \IIIIII rcaacu

.nIhcucw zc—buzas

can
acrpuc
o I ua~» con 50
known I >4 I abso I he
amIautII~u~\ an. IIn I «I II~ —

I IOIuNnI IIn I «I IIugI—I. I anIxu I u I III I arse

u=rupaou con

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w>~p_m0I I II I uzuozquoau uaup .

I

bar—race ccw

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m x -Iqu I .m. I.n I.nII.~. I u I nwao I IIcu
u:a—>zcu

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can

on

 

 

 

5:2-LIICL I.—.Io \

\ LKII.III~3I3‘n!)‘l.:2-..inl ~,.I|QIII'CII..I»I,.IIPIIIIIIIINII'\I~.I

118

 

 

IIIIIIIIIIII
nx~IIquIqu¢09Iucou
cccfiqucoqucou
~IIII~IIIII.~II.IIIII.I.I.III~III~..I»axIII~Iou
IazuIzou
on cI oc
.mmIII.I.IcuIIxIcu.IIIou
n I NIIIIu I .II I.» INII.~. I u I mmxcu IIonu
I \ ~IIIxu I u I
INIII I «I I III.nII~II .III~II.~.I ~IIIII «axe-I IIIIoe

00 6» co aoIthccu up

00

unzuhzcu cm

zequc

Io Ic—e
c I II I o I uauI
o I II I Izwozuuuoz— man»
a I II I hzwczuuuo man»

on or c¢ “IoIheIthup

nIIc—uInIIIcIrIIooIvIIncInIIIaImIIwu I
IIIIIIIIIUI~IIIIuIIII~IUI_II.II—IIIIIOIIIIIII \zzuos n

mecca \ucoa\ w

IrIw

cIIIII..IIIIIuIIIIcIIaIIsaas.oIIIIcIIozouIIIIIucINIxIInnI I

IvesIsuItquuIcIuInIIwaIIIIIIusInc—vmcIacnuc \!I¢<t\ zcaacu

.IIIIIoII zcquzau

cam
zcrhut
«III: I II I OIoo I oc
a I Iccw~nI I «II I ~IIIIU I I I II I u I run I cIoc

o I ua~I.Icu ou

I:2~Izou ccn

 

 

UI_I_IOI I II I IzworIIuoz_ uI—I
war—Ivan (on
IIIIII
«can. I II I Io I II
.oIIIIUIIooxn. zIIIIIIoI IcI
rcahuc .quzIeau an
II I II I «I I.~I I II
III I III I Ic
~I~III I ru I IIII Ioe
~IIIxu I III I III
.III~II~.\ III: I nuII~ I ~IIII~IIII I III
zo~quau quIo an un: can on

u>~p~mos I II I blatantuo wan»

IazuIzou cc—
uocquIaonIocnIoo—v cI ca
I\III~I NIIIIu I u I . InIIwII.~. I «III I ~IIII.. I Iccu
u:2~Ircu cm
zeaIuI
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c I II I o I wIII
o I II I wzuczuswozu wan»
cIII I Izuczquo UI~I
on CI co acIIcIII. I—
uzzuIzcu on
rcrIII
.cIoe

oI oI cu IcIIIIII.orI..o.Iu.~.II

IIIIIIIIIIIIIIIIIIIIIIII.mIIIoInIIwc I
InuIII~IIuIIIIIIuIIII~Iuc—wr.¢¢.1I.IIcwc.mmuI \z:Ic\ n
uccuc \ucoa\ I

. IaIn

couerIacIcuIGIIsIocIJIIIJJoIoIIaIchozcuIvIIIucInIchpr —

 

 

 

VIIoIcIIIIICIquthVIIrcIvIIIcIvIIAc
IrLIIIIIII.IIII-II>II\IwI—fixIllnfilIchszumrtfl

incur

ll ...

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\ IC:\I\

l.)§§~'f\‘

L

‘

IN\LC O

b )- ‘93). 5‘ \IN

0“-

Lulu:

AWL 10°

CL cc

JaIIL AIR
nIckc» I CIQ

I¢IL§II'V.\§

\tIIIIIIIK can.

1319

 

 

 

can
z¢:Iuc
zcrIuc
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I IucuInI III I «I I.~.I—I. I IIIqu I u I III I cIIc
m \ ImI I “III... .
\IocIInII.IIII.IIUII~III~I.I~III. I ~IIIxu I I I ya- I cIIc
o I wanp tea ..0
uzz—Izou can
zI:II¢
xccu I ..c
u:.I.moI I II I IzwczIIuoz. ua.I
I=2.I2cu can
2c;Iuc
mIIc.c I cacao I II I IIIIIcIIIIIo. I ..c
u>.I.moI I II I Izuczunuo uI.I
.:2.I2cu cc.
wocuoInccanchoc.. CI cc
.IIc.uo.Ix..IIIIcI coo~
.waIIaIxIcUIIxICu.IIccu
m \ NIIIIU I III IIM I~II.~. I u I mmch Inaccu
v \ IIIIII I u I .
I~III I «I I III.III~II .III~.I.~.I ~III.I ImxaI. IIIIcu
I=2~Izcu cr
zI:qu
Ic I—ac
o I II I a I uI.I
c I II I Izuczunuoa. uI.I
I I I. . IIIIIIIII II.I

cm a» co «IoIIcIII.u—

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IIIIIIIIII.IIIIIIIIIINIua.wc.¢¢.wIIuccwmImmxI \22Iex I
.ccuc \mcoax I

62I~

 

 

coIIIII.IIIIIIIIIIOIIJIIIJDIIIIIIIcIIoIcuIIIIIUII~IxIIme .
IvLIIsIIIUIINIrIuInIINIIIIIIquIncIvmcIac—.a \aIcqu 7CIacc

amIrI.—pc zcwbczau

czu
zcrIuI
IoIc.c I cue
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I.I I...\IcaIInIIIII.. I I \ NIIIxu I u I II I II I IcIc.c
u:z~Iaou can
zI:Iwc
waIIInoIIcIcIIcIIcuc
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I IOII~MI III I «I II~.I—I. I IIquu I u I III I chse
a \ .~< I “III... .
\IoIIInII.IIII.I.III~III~I.I~III. I NIIIxu I u I III I CIIc
can oI co ..o.Ie.II.I.
o I ua.I IcI c—u
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75;“...
uccu I c.c
u>~IIIOI I II I Izuozunmozn wI—I
unrurzou ecu
20:»uc
Iouw I III
«Iowa I Inoocc I IIIIo I II. I Ic_¢
IIIwIIII~I.I noooc I ~ooec
~I~Ixu I muIIIIIN I ~IIwa I .nIInIIII I .
vIIchIIIIII~I nuIIIIIn I In II“. \IuInIIIIII I IIIII..I~couc
I::~Ixce or.
r¢:III
.«xII I .mIIIo I IIIII. I c.c
on. 6» cc I IoIIoIII.II

u>.I.mcI I II I Ixuczunuc IIII
bra—>200 CO—

 

 

 

litII.

ithIl.

II>\II\

I~CO¢C o

>‘Ch‘.-

QIOIIID I

nIctu I II.
IslII KCOcC
I'IC\-v45l I

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Clhkr1IIlIclIoiIlICRIJIIIIICIOI'lqcIlCleIrtlIhtIssltIINQ

I115IILItUIIK\CILIHQIIIQIIQItIkkIADIIlIIIIC-III Y)!II|\

I.IIII.IIIII II.

)bklisxk

120

 

 

 

 

czu
z¢:Iu¢

QCIqu o «Io

II I III.\ IIIII I IIIIIu I u I I: IIoIo I chcIc
«I I I ~III.IIocu~nI I ~I.I IcIo

. unzIIzcu

.zcquc

I~xIIIIOIIaIOIIavInIc
InIIIIIII..\ InI IIn I ~I IIw
I IoIUIII III I «I I.~.III. I NIIIIu I u I III I cIIe
a \ INI I IIIIII.
\IocuInIIIIIII.IIoII~III~I.I~III. I NIIIII I u I III I cIIe
can OI co IIoIIoIcc.II
o I uaII can qu
wazIIon
z¢:Iu¢
xxcu I III
u>IIIuOI I II I IzwozucuozI uxII
w:zIIIOu
zeaIuc
ooIo I «Ic
IIIIIIII I I II I Iooeo I.~ I IIIIo I IIIII IIN I IIIII. I IcIo
Iuooeo I mIIco I II. I IcI:
IIInIIII~I.I Ioooo I Neon:

~I~cxw I ruoIIIIu I ~Icaxu I

InI¢~¢II< I I

can

can

cow

 

 

IczIII
I~III I IIIIIo I IIIII. I III
cnI CI cc I .oIIcIIIIII

u>~I~noc I In I Izuczuuuo ua—I

I;zIIIcI coI
uocuoIIocIIooquoI. cI cc
I::II2cu cg
«IIIIIIUIUIIIcUIIIou
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NIIIIwIIxIIIIIIIIInIINIIIIIIIIIIIIIuI..InmIIIIIIcu
IazIIzcu cc
cI cI oe
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I I IIIIII I III III I~II.~. I u I nmIII III-c.
m I IIIIIu I u I
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oI oI co IcIIc.II. II
InzIIzcu tr
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c I In I o I “III
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uccue \uccax ~
It.“

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IvoIIIIIIUIIIIcIuInIIwIIIIIIIIIIIoI.I¢IIcI.¢ <IIIIII IcIIc.

InII¢I~Io ttIIL225

 

 

 

FUNCTION BJ

121

122

 

 

.IIIIavcxIIIocunr
NIIxnxvxuu—Icha cs
xxIOOIuIIIuII: co

as c» cc
.oIIII: cm

ocIcoIonaInquuu

a no ~34I> cz~pprm u»:IIcU

cII>mcnc

~I2I~2

oIcu—
bar—prov rd
oorIoorIncarmwbzozyuu on
.nxxIIcoIpmupr In

II c» cc
nxw IIxIIIIo—IIchhnuhr wn
In.~p.~n..rIIu.u_ In

zu:»ur

con c» co ._I»ch. u~
o.— I fit aoIouIzv u— C»
u aoooIMI ur~¢r rm
.nIcnIrw guys. on

Jucwn I c
cIIfil

JUGS. \IOUfiI\ a

IIuIInIuIInusIIIIursccus IcIIcu

svarnmIIIIsnopI—.I>IIuIII>IIIIw¢u»3taou cquw

III: a; vacuuuazs Junnuc ac coupIquuoIIspuIcaIIcI II ozI

 

 

 

czI z~upwoaoc .x In omI_Iumuc mac—:xuu» zo_»I4II ugzucaauua

ccxpu:

uzcr

cu¢_aouc vaIccccscam acupuznu czI nu:.»:ca¢:v

nu IIz» IUIIuce I con «\xIco

r” o» IIaow co IIII Inwa I «on "I“ IIquIc_Ic~
III» awn;
II pen: I. »:I .oIIN o» IIaou Io zIx» cuprIc II Inna 2

oxaIIuc

ImquauI nun. pIquou pa: - a» cucIIxou 2 ac uoaII IIIU.
aux—Isa: pox IUIcauUI ouc—aouc nIcu—
c¢u~ I: u>_»Ieuz n— I «III.
u>ppqur v» z uIcw—
chIu c2 oIIu~
uuuxI ucou IaIIu I2I»42¢~IIIU.
IuIc:qu cac—aouII c
zo~»u:=u JIIIII a I2I.4:IUI IIII II
ouc—muc II—Iulau quanta 1 ya» u: Iuccc upr 2
egg—vac Io.»uz:u Junruc a Us» I: IzuaauII upr I

IIUIUIIIII so :c.»I_IumuI

act—IoIfilIrI-gfiwuc JJIU

uccn:

IucIo ozI pauaach III—e I up. Io.»uaau dunnuc 5 us» upaIIcu
uncccaI

anuc wrapaccc:¢

IIIxufio Io~>utak

U

009

 

 

 

123

 

 

 

czw

In. I Iuoco o~4I>z~ zoupuzns JumnuIJOI I x.I»Ianu 0.9

 

 

”won‘t I

u>~>Iauz mp pzua:ch zoubuzak Juwmwnxsn I x—IhIaccu

Icpm

z Iago.n.u»_cx

zczwuc

I— I new: I nah:

nIcu—

III>u¢II
oo—.oc~.oo~n.wmIOIIIIIn>ucIcIac.ncIII—
. IxchxficIfio
xchIxIIIIszJI

xchac

oc—IosuICI—azuu—
IIIIIIIIIIwaxc
mechIxIIIIIIIJI

pr'Iv

pfiIIhfi

xxaIfic

cm—ICI—Icn—anoznr9~u~
racuplu

IIIIII
auoxx«auIszthcakIIwaxc

XIIIII

«II—Ix cc. a?

N's-N:
«Irfi

On— c» be
uIIhfi

ow—Io—qua—nmIawxauuauu—

cIIIaIII
cIIab

IIIIc._I_II

I—IavuIatvb >uv

nI-IIIIocu~III co. cc

unuthuIaI

coo

con

oo—

cs—

cc—

on—

en—

en“
emu

en—

ce—

 

 

 

SUB-ROUTINE TIMER

124

125

 

czu

2¢=Iuc

term

III—IIIII_»II~uI_»Inua~In .oIIn. yuan:

IIcOIan—uuocaIIruaup

coxnuaupInua—px

.IoonwxupvocaIIwuaup

cox~uI~II~uauwx

Icoxnnuux—wInuaup

Icexawuux~uI~wI~I

IImcvouwv IonuIImurazua InuIIIIIIIIIc~ cuxcnoaouum InIcuII_
mu»az.a II_I...«mIoc¢I IIIIIII ouaavzou ma.» .ocw I: czUII. IIIIcI c.

cuxuh uz_»:ccc:e

 

APPENDIX B

DETERMINING MATERIAL PROPERTIES

B-l Poro-elastic Material Constants

 

In order to establish the validity of Eq. (36) of 3. 4, set the p equal to zero

in (31) and solve for 7 . This gives Eq. (1):
a
VI 3% 0mm (if 9'0) (1)

which relates 0mm to 7 for the case of zero pore pressure. Differentiating

Eq. (1) with respect to time yields the rate Eq. (2).
._ =-__ _ (if pm (2)

Substituting Eq. (24) of 3. 3 for -g—-'Y- in Eq. (2) yieldsathe following result.
I
m:-&; ._"‘.£'. (pr30) (3)
3' 06m“) 3'
If the constraint condition Eq. (26) of 3. 3 is substituted in Eq. (3) and both

 

as
sides of the resulting equation are divided by any“ , the desired result is

obtained.

 

as
decaf, ' , 0 (4)

Equation (4) was derived under the condition of zero pore pressure; however,

0' .-

since this equation contains only constants it follows that it must be true gen-
erally.

By considering the definition of the constants 0| , and c , and the con-
stitutive equations it appears that six independent material constants determine
the mechanical properties of the poro-elastic mixture:

(1) R , the pore compressibility

(11) f , the porosity
126

127

(iii) 02 , the shear modulus

(iv) 03 , the apparent Lame’constant at steady state

(v) (04127508) , apparent Lame’ constant at time zero for total stress
tensor .

(vi) a , the diffusion constant.

However, for the material being considered in this problem, (v) can be expressed
in terms of (1), (iii), and (iv) as shown in Eq. (5)

.2. g a -
3024-04 p08, (302+03)/(l R) (5)

The validity of Eq. (5)2 can be established by considering physical tests
proposed by Biot and Willis [1] . Of the tests mentioned in [1] the "jacketed
compressibility" test is of primary interest. In the jacketed compressibility
test, a sample of the poro-elastic material is enclosed in a thin impermeable
jacket and then subjected to an external fluid pressure p' . The inside of the
jacket is vented to the atmosphere so that the excess pore pressure is main-
tained at a negligible level in the sample. Although this test can be performed
on a dry sample, the data obtained are more useful if the test is performed on
a fluid saturated sample. After external pressure is applied to the sample,
the specimens begins to compress and water flows out from the pores. Even-
tually a steady statecondition is reached when the volume of the specimen
stops changing and the fluid flow ceases. When the steady state condition is
reached the total sample volume change and total pore water outflow is meas-
ured. From this measured information it is possible to compute I: the coef-

ficient of jacketed compressibility which is defined by Eq. (6).

0mm
4' ' ' p. (6)

Substituting this expression in the constitutive equations (34) of 3. 4, and noting

 

that 1r” is zero in this experiment, leads to Eqs. (7 ) and (8),

128

(Z)-

I. p. :- 2502‘ p'-a4kp'-08 pvm’m

(7)
v .- aekp'
mtm 06(215 (8)
Combining Eqs. (7) and (8) yields Eq. (9):
l 2 ‘1
;=§az+a4-a%8-§-oz+oa (9)

which may be used to determine (:3 if the shear modulus 02 is known.
Knowledge of the amount of water forced out of the sample during the

jacketed compressibility test can be used to determine the constant 0' . It

is observed that the measured water outflow per unit sample volume § is given

by Eq. (10).
£‘f (.mm'Vm’m) (10)

1)in1118 EC!- (10) by 'mmv the measured change in sample volume per unit volume,

noting that ”ijIO and using the second Eq. (34) of Chapter III, provides Eq. (11).

C f 0'
mIm—(Imm-vm’ml-(-‘-2$%;+I)f-(l+o|lf (11)

Since both 5 and Imm are measured in the jacketed compressibility test, it
follows that if the porosity f is known, the constant a. can be determined
using Eq. (11).

In order to show how the constant §Oz+a4 is related to a. and .1: , it is
convenient to introduce the unjacketed compressibility test although
these test data are not actually required to determine the desired con-
stant. In the unjacketed compressibility test , the sample of fluid filled mate-
rial is immersed in a container filled with the fluid component of the mixture
and a hydrostatic pressure p' is applied to the fluid surrounding the sample.

The dilatation, or total volume change of the sample from its initial unit value

129
to that in steady state, is measured in this test and is denoted 5 . It is noted

inthis test,that the partial solid stress normal to the sample surface is equal
to -(l-f)p' and that the partial fluid stress, 1r“, , is equal to -fp'8m . [1]

The unjacketed compressibility constant is defined by Eq. (13).

6--"“T.'“ (13)
Using Eq. (13) in Eq. (34) of Chapter III leads to Eq. (14).
2
2 (a )
l-(l+o')f-[§oz+a4-—%J8 (14)

Combining this result with that given in Eq. (9) and solving for i in terms of

“I and i provides Eq. (15). I .

3' Hl-(Hallf) (15)

In the unjacketed compressibility test so , as the fluid in the pores is

' 'm,m
incompressible and the test is run under undrained conditions. Therefbre, the

second Eq. (34) of Chapter III yields an expression for the constant imbue in

terms of 8 which can be used as shown in Eqs. (16) and (17):

 

 

-m-a ,'_P_ L _ (16)
p 8 9mm: b
2
(a )
2 9 .0'1 (17)
(03) 06 5
to compute .
“6
Substituting Eq. (17) in Eq. (9) and solving for gee-+04 yields Eq. (18).
' 2 f l
3024-0480'34-2 (18)

Finally, substituting Eq. (15) into Eq. (18) yields an expression for the con-

stant -2- 02+a4 which can be computed using the "jacketed compressibility"

 

 

 

3
test data.
2 . (l-f)
3 °2+°4 k(I-(I+o,)n (19)
Substituting Eq. (15) in Eq. (16) gives expression (20) for 42’5 as .
-<z)- . f
”as ku-(Hapn (20)
Adding Eq. (20) to Eq. (19) provides another useful expression.
' 2 <2:- I (21)
_ +a - a s
302 4 p 8 Ail-(Ham)

130
Using the definition of °l given'by Eq. (26) in 3- 3. EQ- (21) can be reduced to (5)

of this section. The definition of 0' and Eq. (11) yields Eq. (22):

R8 —c (Jacketed Compressibility Test) (22)

0mm
which can be used to compute the pore compressibility constant, R ,using .
jacketed compressibility test data. Equation (22) shows that the pore com-
pressibility constant R is approximately equal to the ratio of water squeezed

out to the total sample volume change occurring in the jacketed. compres—

sibility test. Equation (23):
a. “'“mmlifll .(23)

0mm
is correct but would differ from (22) only slightly if the measured dilata-

tion 9mm is small compared to unity. Equation (22) is consistant with the
small strain assumptions adopted for this study.

The porosity constant, f can be obtained by first determining the specific
gravity, 8.6. , of the skeletal material contained in a sample of the poro-elastic
mixture. The porosity, or Void space per unit volume of the skeletal
material can be computed from the specific gravity information as shown in

Eq. (24) .
_ (Sample Wain/II (dry) / Unit Val.)
Density of Water

f" /S.G. (24)

The specific gravity of soil materials.which are of primary interest in this
study, should be determined in accordance with ASTM Specification D 854-58
(AASHO No. T100) .

The diffusion constant a is related to the Darcy coefficient of perme-

ability, D.C. P , by Eq. (25):
o . fluid donsily

(25)
(D.C.RH

 

131
where the fluid density is that of the fluid contained in the pores. The perme-

ability constants for granular soils should be determined according to ASTM

D 2434-68.

Finally, it is necessary to determine the shear modulus 02 . (be way
of determining this constant is to superimpose a unixial stress in addition to
the hydrostatic pressure p' at the end of the jacketed compressibility test.
Assuming the additional loading is applied in the z direction, 02 can be
determined from Eq. (26):

(A022 -03 A 0mm)

02 . ZAOII (26)

 

which follows when p is set equal to zero in the first Eq. (37) of Chapter III.
The axial strainngnin Eq. (26) can be determined by measuring the axial

deformation of the sample.

132
B—2 Stress Relaxation of Sand-Asphalt Mixtures

 

The results of stress relaxation measurements on a sand—asphalt mixture
that were obtained by Moavenzadeh and Soussou [E] are used in the example
problem that is presented in Chapter X of this report. These particular data
were chosen because of the convenient mathematical form in which they were
presented. It would be more physically realistic to utilize data from tests on
bituminws concrete as flexible pavements are constructed primarily of
such bituminous and coarse aggregate mixtures. With regard to this,it is noted
that the quasi-static response of sand-asphalt mixtures and of bituminous con-
crete has been studied by several other investigators in recent years [22, 2; ,
a] . The data from tests on these other mixtures could be put in the same
form as that given in reference [Q] by using a numerical collocation technique
that will be described here.

The sand-asphalt mixture used in the Moavenzadeh study was made up of
Ottawa sand and asphalt cement. The gradation of the Ottawa sand used is

given in Table 1.

TABLE 1
GRADATION OF AGGREGATES

Percent Passing*

 

 

 

Sieve Size
I ASTM D 1663-59T Selected Gradation
16 85-100 100
30 70- 95 75
50 45- 75 45
100 20- 40 26
200 9— 20 15

 

* Percent of the total weight of material passing a
given sieve size.

133
The asphalt cement used was an AC-20 grade asphalt and the results of

some conventional tests on the material are given in Table 2.

TAB LE 2
RESULTS OF TESTS ON ASPHALT

 

 

 

Test Result
Specific gravity, 77/77 F 1.020
Penetration, 200 gm, 60 sec, 39.9 30
Ductility, 77 F 250 + CM
Flash point, Cleveland open cup 545 F

 

Relaxation test data were obtained for the mixture by subjecting it to
uniaxial compressive strain. Corrections were introduced to account for the
fact that the initial loading near time zero was not constant but was increased
from zero to the desired constant level linearly with time. An exponential series
function of the form Eq. (27): j
. ¢(')=¢(°°)+if| Bio-bit (27)
was fitted to the stress response versus time data using a technique developed
by Schapery [2_3]. Basically the technique consists of selecting the b; coef-
ficients to span several decades of time, and then selecting the a. to mini-
mize the mean square error between the function and the data.

A ten term series was selected to model the sand asphalt mixture, with

the bi coefficients defined by (28).

bi=5*lo4xlo". i-I low (28)

Fitting the curve to the data by the least squares technique yielded the Bi

coefficients as indicated in Table 3.

134

TABLE 3

NUMERICAL VALUES OF COEFFICIENTS
OF STRESS RELAXATION FUNCTION

 

¢(t)=¢(°°)+2 sio'

IO

bi'

 

where:

6.65 x 103
2.32 x 10‘
4.19 x 10“
3.30 x 10*
3.01 x 10‘
1.26x 10‘
4.30 x 103
1.44 x 103
8.36 x 102
4.03x 102
1.60x 103

5x103
5x 103
5x 101
5x 10"
5x 10"1
5x 10'3
5x 10'3
5x 10“
5x 10"
5x 10'6

 

APPENDIX C

INITIAL AND STEADY STATE SOLUTIONS

The solutions corresponding to the Laplace-Hankel transformed solution
images can be readily determined for the initial state and steady state [24:] by

computing the limits shown in Eq. (1).

 

9v(|\.l) 8 lim s§y(q,s) . . initial value can
'30 fi-rc
(1)
{It/(mill - lim shims) , shady stale can
"a, 8+0

The initial value and steady state transforms can then be inverted numerically
by approximating the inverse Hankel transformation operation shown in Eq. (2).

HM), - lim gv(q,0)qu(q,r)dq (2)

t- on a"
The limits indicated in Eq. (1) have been analytically determined for the
double transforms that were obtained in Chapter VIII. The limit functions ob-
tained are presented here.
The initial value limiting functions can be predicted independently because
the initial values of all the dependent variables can be obtained in the same

manner that the initial value of the dilatation was determined in 7. 2 - The

initial values of the Hankel transforms of the dependent variables obtained

using limiting process Eq. (1) are given in Eq. (3).

135

136

I‘m 356(0,” 3 -k0(02 + a4-W03)

 

"I_L"Q3fio('lr') - 2 q (202+a4-b'mae)

 

. - kc (202 + a4 -5"-’03)(| + '12) kc: .
I I ’— _ __ “I
.1"..”zo"""" I: 2020 2 °

,'§L"..“mmo(q'z") = -|u:c"'\z

Iim sPom, z,s)-I :32: )(- 03+a4- Pmael

‘-.Q

_ - z -

scum

 

+(202 + 303)]
‘mstammo(n,z,s) -330(n,z,sl)- -kce"‘z(202+ 3(04-5"’a3))

llm 86120 (I1. Z,8)"k00 m[qz(02+a4-D"’03)

s —>cu
+(03 + 0|(04-Pm03))+ a]
('4' 0|) 02

 

’ Liradénoimzdl-Pom .z.s)) = -kc€"z(az+ a4-5“'ael(qz +I)

 

s Taunma, sl=

29:12“[kc(02+a4- 5‘3’03) nz- Rena]

. - '_ -72) z
‘lflaan'hlflfil chtaz+a4 p aa)0.n Z

sd5'
"m 711(n,2,8)

kw dz . -qo"l‘kc(02 +04-5m03Nl-qz)

 

Iim SIC-THO -2 P0 4‘ 0990 I ‘ «can: [02 4' 2‘04- -p“’aa I

s-o-oo

+ qz (02 + 04 - P‘z’aa )]

(a)

(b)

(C)

(d)

(e)

(f)

(8‘)

(3)

(h)

(i)

(j)

(k)

(1)

(m)

137

Using Terezawa's solution [12] the initial condition defining problem of 7. 2

was solved and expressions Eq. (3a), (b), (d), (g), (i), (j), and (k) were ob-
tained by this alternative procedure. These results verify the validity of the
computed limits indicatedsililxpression Eq. (3e), (f), and (h), are either used

in combination to make up the other expressions ,or are made up of a com-
bination of the others. Therefore, the results obtained by direct solution

imply that Eq. (3e), (f), and (h) are valid also. Examining Eq. (3a), (b), (c),
(e), (h), and (k),it can be seen that the boundary conditions at z = o are satisfied

at i=- 0 as summarized in Eq. (4a), (b), and (c).

 

s'flgfliflfl.“ ' s'fl;(ozzo("")'P°(""n (a)
z=0
smsiomfi) = £12.98?!“th z :0 (b) (4)

 

= O
z 8 O
The steady state Hankel transforms of the dependent variables are given

8 lim 8 Cram,” (0)

 

by Eqs. (5) and (6).
qul(bq)202

K a -r (5)
[0R (on)q4(2a2+ o3)+ 2an (a2 + 03)] c

 

l' a" , )--Kc(a +0)
’20 “0"!“ 2 3 (3)

(2024-03)

Homgm”) ' 2020

' (b) (6)

. . '01 Kc(202+o§X|+'\1) _ kcz
.mosozomg,” o [ 202“ --—2 (C)

 

“I The plate deflection and foundation reaction at time zero, given by
Eqs. 3a and 3b, also agrees with the known solution for an elastic
plate supported on an elastic half-space and subjected to a circular
uniform load [25]..

138

. ._ w ' d
.mosimmo (11.2.3) Kc. ( )
I'm sPI ,z,s) 8 O (8)
’LO 0 'l
- ‘ . . -- 2 + ’"‘ f
. .mosammOMIS) ( 02 303M" ()

8920' 6110 (n.2,s) - 00"“ [kozqz -K (02 + 03i-K(202+03)nz:| “9

 

 

lim :6 ( z s)= in: M20 +0 ) z-Kco -kca 2 (h)
g-po r1 '1' ' 202“ 2 3 '1 2 2'1 (6)
£130: CJ'Irzl (q,z,s) . kcozo'“: nz - Kc(202 + 03) 0"“ '12 (i)

n 3 a 4110‘ 4. |.. -

s-E-‘O dz no C(202 03)( n2) K002 (j)

-i- kcaz 01]

. - _ - . _ 412
£120 . (0"0 29° + 0000) Kc(02 + 203k (k)
+ Kc (202 + (:3) Iain: ‘kCOgnzeqlz

Examining Eq. (63), (e), and (g) verifies that boundary condition , Eq. (7a), is
satisfied. Using Eq. (6b) and (c) with 2 set at zero, it can be seen that Eq. (7b)

is satisfiedalso. Setting z equal to zero in Eq. (6i) shows that b’oundary con-

dition Eq. (7c) is satisfied in the steady state.

(a)

lim :5” ,0)- lim 5(5 (mud-P (11.1.3))
3*0 O '1 8"0 22° 0

 

1'0

limos 53 (11,3) 311.20: 0,001.2.”

,_, 0» <7)

 

280

=0
:80

11208 0(1'1n010’) (C)

 

   

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