Masters in Earthquake Engineering
and Engineering Seismology
BASIC TRANSFER AND TIME-RESPONSE
FUNCTIONS OF THE THREE PARAMETER
FLUID AND SOLID VISCOELASTIC MODEL
by
KAMPAS GEORGIOS
Supervisor: PROFESSOR NICOS MAKRIS
DEPARTMENT OF CIVIL ENGINEERING
UNIVERSITY OF PATRAS, GREECE
October, 2007
Abstract
ABSTRACT
In this thesis, the basic transfer functions and time response functions of the three
parameter fluid and solid models (Jeffreys’ and Poynting-Thomson’s model) are
revisited. The two models find application in several areas of engineering and
geophysics, including in the modeling of the behavior of earth strata. The relation
between the analyticity of the transfer functions and the causality of the corresponding
time response functions is established by identifying all singularities at ω=0 after
applying the partial fraction expansion method. The strong singularity at ω=0 in the
imaginary part of the transfer functions in association with the causality requirement
imposes the addition of a Dirac delta function in their real part. This operation makes
possible the application of time domain techniques that do not suffer from violating
the premise of causality.
i
Acknowledgements
ACKNOWLEDGEMENTS
I would like to thank Professor Nicos Makris for introducing me to the topic of
viscoelasticity and linear viscoelastic models, for the proper guidance with his
expertise, for his enthusiasm and motivation, but most of all for his helpful
inspirations during this work.
I would like also to thank the program of Master of Earthquake
Engineering and Engineering Seismology (MEEES) course and the European
Commission for the financial support during the one year of my studies.
Finally I would like to thank my family for the unconditional support and my
colleagues in this Master course for sharing with me all the wonderful experiences
that evolved us to better engineers and made us citizens of the world.
ii
Index
TABLE OF CONTENTS
Page
ABSTRACT………………………………………………………………………….................i
ACKNOWLEDGEMENTS…………………………………………………………................ii
TABLE OF CONTENTS……………………………………………………………………..iii
LIST OF FIGURES…………………………………………………………………...............iv
LIST OF TABLES……………………………………………………………………………..v
1. INTRODUCTION…………………………………………………………………………1
2. FUNDAMENTAL OF LINEAR VISCOELASTIC THEORY AND MODELS…………3
2.1. Introduction……………………………………………………………………….3
2.2. Viscoelastic models……………………………………………………………....3
2.3. Analogies between rheology and structural Dynamics..………………………....7
3. TRANSFER AND TIME DOMAIN FUNCTIONS…………………………….……….10
3.1. Analyticity and Causality……………………………………………………….10
3.2. Strictly proper/causal functions…………………………………………………12
4. REVISION OF THE THREE PARAMETER SOLID ...………………………………..14
4.1. Dynamic stiffness and memory function……………………………………….14
4.2. Dynamic flexibility and impulse response function……………………………15
4.3. Impedance and relaxation modulus…………………………………………….16
4.4. Mobility and step response function…………………………………………....16
4.5. Plots of the transfer and time domain functions………………………………..18
4.6. Several comments on Poynting-Thomson’s model..……………………………19
5. REVISION OF THE THREE PARAMETER FLUID .………………………………….21
5.1. Proof of the constitutive equation……………………………………………….21
5.2. Dynamic stiffness and memory function………………………………………..22
5.3. Dynamic flexibility and impulse response function…………………………….23
5.4. Impedance and relaxation modulus……………………………………………..24
5.5. Mobility and step response function…………………………………………….25
5.6. Plots of the transfer and time domain functions………………………………...26
5.7. Several comments on Jeffreys’ model…………………………………………..28
6. ADMISSIBILITY CRITERIA FOR THE RELAXATION MODULUS.……………….29
7. CONCLUSIONS…………………………………………………………………………31
8. REFERENCES…………………………………………………………………………...32
iii
Index
LIST OF FIGURES
Figure 3.1. Reciprocal function………………………………………………………………11
Figure 4.1. The Poynting-Thomson’s’ model ……………………………………………….14
Figure 4.2. Plots of the time domain functions of the Poynting-Thomson’s model…………19
Figure 4.3. Plots of the frequency domain functions of the Poynting-Thomson’s model…....19
Figure 5.1. The Jeffreys’ model………………………………………………………………21
Figure 5.2. Plots of the time domain functions of the Jeffreys’ model ………………………27
Figure 5.3. Plots of the frequency domain functions of the Jeffreys’ model …………...……27
iv
Index
LIST OF TABLES
Table 2.1. Models’ physical representation……………………………………………6
Table 2.2. Structural dynamics’ time response functions……………………………..7
Table 2.3. Structural dynamics’ frequency response functions……………………….7
Table 2.4. Rheology’s time response functions……………………………………….7
Table 2.5. Rheology’s frequency response functions…………………………………8
Table 2.6. Famous rheology experiments……………………………………………..8
Table 2.7. Rheology versus structural dynamics……………………………………...9
Table 4.1.Transfer and time domain functions for Poynting-Thomson’s model ...….18
Table 5.1. Transfer and time domain functions for Jeffreys’ model…………... ……26
v
Introduction
1. INTRODUCTION
The linear theory of viscoelasticity is traditionally used by a wide class of
scientists and engineers. Its main purpose is to model and predict the response of
engineering systems that do not exhibit purely elastic behavior -as most systems work
in nature. Disciplines that take into advantage already the viscoelastic models is
engineering seismology, rheology, structural mechanics, dynamics etc. Thus it is for a
multidisciplinary benefit to investigate the fundamental properties of the viscoelastic
models. The aforementioned models are consisted from elements like the “Hookean”
spring and the “viscous” dashpot that are linearly superimposed creating that way
phenomenological viscoelastic models.
Furthermore, for finding the response of any viscoelastic model for different
excitations one should use either the frequency domain or time domain techniques. It
was found that although the frequency domain techniques can be used without any
problem, the time domain techniques had a severe problem; causality requirements in
the time domain functions of the models were not satisfied. Although the model was
working successfully for engineering purposes, it was not well defined from a
mathematical and -more importantly- from a physical point of view as the system was
responding even before the excitation.
The investigation has started with Makris (1996) in fundamental viscoelastic
models like the “Hookean solid” and “Newtonian fluid” or Kelvin-Voigt and Maxwell
model, in which the relationship between the analyticity of the frequency domain
transfer functions with the causality of their time domain functions was applied. The
result of this was that the real and the imaginary part of the transfer functions have to
be Hilbert pairs or to satisfy the Kramers-Kronig relationships in order for the
resulting time domain functions to be causal.
A continuation of this investigation is presented in this thesis in a bit more
complicated models used by seismologists [Jeffreys (1924)] and engineers. The three
parametric fluid and solid model – or else called Jeffreys’ and Poynting-Thomson’s
model- were introduced and subjected to an analogous procedure. In several cases the
transfer functions had an extra “hidden” pole in zero in the imaginary part, which was
revealed using the partial fraction expansion method, without having an appropriate
Hilbert pair in the real part. According to the previous ascertainment a Dirac delta
function had to be applied to the real part thus concluding to a causal time domain
1
Introduction
function. Additionally in Chapter 2 the fundamentals of viscoelasticity are presented
in a more solid way, dealing also with the various models. Furthermore, several
conclusions had been made by the observation of the resulting time domain functions
and their plots. The commenting was focused more in the relaxation modulus as it is
the most important function of the models since it displays the response of them in
any excitation. Several other admissibility criteria were investigated resulting in
useful results.
2
Fundamentals of linear viscoelastic theory and models
2. FUNDAMENTALS OF LINEAR VISCOELASTIC THEORY
AND MODELS
2.1 Introduction
The linear theory of viscoelasticity is traditionally used from a large amount of
scientists and engineers. Its main purpose is to model and predict the response of
engineering systems that do not exhibit elastic behavior -as most systems work in
nature. Wide application of the linear theory of viscoelasticity can be seen in
disciplines like rheology, structural engineering as dissipation systems and structural
dynamics. The big advantage of the theory is that it evolves an inductive manner
starting form the “elastic spring” (Hookean Solid) and the “viscous dashpot”
(Newtonian fluid) and proceeds to more comprehensive phenomenological models by
linear combinations of the two aforementioned basic elements.
Thus, the linear modeling of the “elastic spring” and the “viscous dashpot”
leads to constitutive equations with constant coefficients of the form
M
[ ∑α m
m =0
N
dm
dn
P
t
b
]
(
)
=
[
]u (t )
∑
n
dt m
dt n
n =0
,
(2.1)
where P (t ) and u(t ) are the time histories of the force and the small-gradient
displacement;
α m and bn are restricted to real numbers and are the parameters of the
constitutive model and the order of differentiation m and n is restricted to integers.
2.2. Viscoelastic models
Specific models have been developed dependent on the system each time. Famous
examples of those models are:
•
Hook’s model
Together with the “Newtonian dashpot” is the most fundamental element in
viscoelasticity. It represents a fully elastic solid and it is symbolized with a spring.
Its constitutive equation is
P (t ) = Ku(t ) ,
(2.2)
where K is the stiffness of the spring.
3
Fundamentals of linear viscoelastic theory and models
•
Newton’s model
Newton’s model is the famous dashpot. It represents a fluid and it is usually used
for modeling the dissipation elements of a system. It is symbolized with a dashpot.
Its constitutive equation is
P (t ) = C
du(t )
dt
(2.3)
where C is the damping coefficient or the viscosity of the system.
•
Kelvin-Voigt’s model
It is the most popular model as far as it concerns the modeling of a structure in
structural dynamics. It represents an inelastic solid (or structure with damping). It
can be symbolized with a spring in parallel with a dashpot, thus assuming that the
two elements have common displacements but they “share” the forces. Its
constitutive equation is
P (t ) = Ku(t ) + C
•
du(t )
dt
(2.4)
Maxwell’s model
It represents a fluid with viscoelastic behavior. It is symbolized with a spring in
series with a dashpot, thus assuming that the two elements have common forces
but they “share” the displacements. Its constitutive equation is [Makris (1996)]
P (t ) + λ
where
•
λ=
dP(t )
du(t )
=C
dt
dt
(2.5)
C
.
K
Poynting-Thomson’s model
It is a three parametric model that represents a more complicated behavior of a
solid. It can be symbolized as a spring element in series with a Kelvin-Voigt
element. Consequently the spring “shares” the displacement with the Kelvin-Voigt
model and they have the same force. Its constitutive equation is
P (t ) + λ1
where
dP(t )
du (t )
= K [u (t ) + λ2
]
dt
dt
(2.6)
λ1 is called the relaxation time and λ2 the retardation time.
4
Fundamentals of linear viscoelastic theory and models
•
Jeffreys’ model
It is a three parametric model that represents a more complicated behavior of a
fluid and it can be used in modeling the earth as the propagation path of P and S
waves in the discipline of engineering seismology. It can be symbolized as a
dashpot element in series with a Kelvin-Voigt element. Consequently the dashpot
“shares” the displacement with the Kelvin-Voigt model and they have the same
force. Its constitutive equation is
dP(t )
du(t )
d 2 u (t )
P (t ) + λ1
= η[
+ λ2
]
dt
dt
dt 2
where
(2.6)
λ1 is called the relaxation time, λ2 the retardation time and η is the
viscosity of the liquid. But as it was aforementioned in previous chapters it is the
target of this thesis to present detailed in another chapter Jeffreys and PoyntingThomson’s model since they are more involved in engineering seismology.
5
Fundamentals of linear viscoelastic theory and models
Hook’s model
Newton’s model
Kelvin-Voigt’s model
Maxwell’ model
Table 2.1 Models’ physical representation
6
Fundamentals of linear viscoelastic theory and models
2.3. Analogies between Rheology and Structural Dynamics
The linearity of those models allows using both time domain and frequency
domain analysis for small magnitude displacements/strains. The expected operation in
time domain is convolution and in frequency domain is multiplication. Using equation
(2.1) using the Fourier transform all functions in time and frequency domain can be
deduced. Generally it can be defined:
Memory Function
q(t )
P (t ) = q(t ) * u (t )
Impulse Response Function
h(t )
u( t ) = h (t ) * P (t )
•
Relaxation Modulus
k (t )
P (t ) = k ( t ) * u (t )
Step Response Function
m(t )
u = m(t ) * P ( t )
•
Table 2.2 Structural dynamics’ time response functions
Dynamic Stiffness
K (ω )
P(ω ) = K (ω )u (ω )
Dynamic Flexibility
H (ω )
u (ω ) = H (ω ) P (ω )
Impedance
Z (ω )
P(ω ) = Z (ω ) u (ω )
Mobility Function
M (ω )
u (ω ) = M (ω ) P (ω )
•
•
Table 2.3 Structural dynamics’ frequency response functions
Alternatively tables 2.4, 2.5 can be defined from Giesekus (1965)
Retardation Fluidity
φ (t )
γ (t ) = φ (t ) * s ( t )
•
Relaxation Modulus G (t )
s( t ) = G ( t ) * γ (t )
•
J (t )
γ (t ) = G (t ) * s (t )
Stressing Viscosity η (t )
s( t ) = η (t ) * γ (t )
Creep Compliance
••
Table 2.4 Rheology’s time response functions
7
Fundamentals of linear viscoelastic theory and models
Complex Fluidity
φ(ω )
γ (ω ) = φ(ω ) s(ω )
Complex Modulus
G (ω )
Complex Compliance
J (ω )
γ (ω ) = J (ω ) s(ω )
Complex Viscosity
η(ω )
s (ω ) = η(ω ) γ (ω )
•
s (ω ) = G (ω ) γ (ω )
•
••
Table 2.5 Rheology’s frequency response functions
where the symbol “ * ” symbolizes the convolution operator. According to
Giesekus (1965) it is obvious (using the Fourier transform) that
G (t ) =
dη (t )
dJ (t )
and φ (t ) =
that is, the relaxation modulus is the
dt
dt
derivative of the stressing viscosity and the retardation fluidity is the
derivative of the creep compliance.
However in the discipline of rheology there are several experiments that
because of the very specific input the convolution relationships tend to be
multiplications in the time domain as it seems in Table 2.6.
EXPERIMENT
INPUT
OUTPUT
Creep Recovery Experiment
s(t ) = s0δ (t )
γ ( t ) = φ ( t ) s0
Relaxation Experiment
γ (t ) = γ 0 δ (t )
s( t ) = G (t ) γ 0
Creep Experiment
s ( t ) = s0 H ( t )
γ (t ) = J (t ) s0
Stressing Experiment
γ (t ) = γ 0 H (t )
•
•
•
•
s( t ) = J (t ) γ 0
Table 2.6 Typical rheology experiments
8
Fundamentals of linear viscoelastic theory and models
Additionally it is clear the correspondence of structural dynamics’ and
rheology’s time domain functions. The relaxation modulus for both disciplines is
similar k (t ) ↔ G (t ) and the impulse response function with the retardation fluidity
h(t ) ↔ φ (t ) .
In structural dynamics it can also be stated that
•
•
P(t ) = k (t ) * u (t ) ⇒ F[ P (t )] = F[ k (t ) * u (t )] ⇒ P (ω ) = Z (ω )iω u (ω )
(2.7)
P(t ) = q (t ) * u (t ) ⇒ F[ P (t )] = F[q (t ) * u (t )] ⇒ P (ω ) = K (ω ) u (ω )
(2.8)
The division of (2.7) with (2.8) leads to
K (ω ) = iωZ (ω ) ⇒ F −1[K (ω )] = F −1[iωZ (ω )] ⇒ q (t ) =
dk (t )
dt
(2.9)
And by following the same pattern
m( t ) =
dh(t )
dt
(2.10)
that is, the memory function is the derivative of the relaxation modulus and the
step response function is the derivative of the impulse response function.
Using (2.9) and (2.10) it can be deduced that q(t ) ↔
d 2η (t )
and
dt 2
d 2 J (t )
m( t ) ↔
.
dt 2
To sum up table 2.7 shows all the above analogies between rheology and
structural dynamics.
STRUCTURAL DYNAMICS RHEOLOGY CORRESPONDENCE
q( t )
η (t )
d 2η (t )
q(t ) ↔
dt 2
h(t )
φ (t )
h(t ) ↔ φ (t )
k (t )
G (t )
k (t ) ↔ G (t )
m( t )
J (t )
d 2 J (t )
m( t ) ↔
dt 2
Table 2.7 Analogies between rheology and structural dynamics
9
Transfer and time domain functions
3. TRANSFER AND TIME DOMAIN FUNCTIONS
Linear viscoelastic models, although they work very well for practical
engineering purposes, they were physically not well defined. They had not been
fulfilling the causality criteria since in Makris (1996) the time and frequency domain
functions were re-defined correctly for Hook’s, Newton’s, Kelvin’s and Maxwell’s
model. In this thesis the main purpose is to define correctly –by fulfilling the causality
criterion- the time domain and transfer functions for Jeffreys’ model since it is widely
used in Seismology. The same results are obtained also for Poynting-Thomson’s
model, since it is the three parametric solid model, for comparison purposes.
3.1 Analyticity and Causality
The initial criteria for a time domain function are to be real and causal.
However the transfer functions in the frequency domain are complex functions.
Consequently, as t < 0 the transfer functions should be analytic in the right-hand
complex plane [Papoulis (1987)], and the term analytic means that the transfer
functions are complex differentiable in the right-hand complex plane. This condition
on the analyticity of the transfer functions dictates that their real and imaginary part
should be related by the Hilbert transform [Makris (1996)]:
Re[ f (ω )] = −
1 +∞ Re[ f ( x)]
1 +∞ Im[ f ( x )]
ω
dx
,
Im[
f
(
)]
=
dx
π −∫∞ x − ω
π −∫∞ x − ω
(3.1)
The proof of the relations given by (3.1) can be found in text-books [Bendat and
Piersol (1986); Papoulis (1987)] as it is not the purpose of this thesis to present it.
They can be derived directly from the Cauchy integral theorem and are known in
viscoelasticity as the Kramers-Kronig relations [Booij and Thoone (1982)]. Thus for
the time domain function to be causal, the real and imaginary part of its Fourier
transform (transfer functions) should be Hilbert pairs or equivalently satisfy the
Kramers-Kronig relationships.
As an example, it can be shown that the real and imaginary parts of the
complex conjugate of the Dirac delta function are Hilbert pairs [Makris et al (1996)]:
~
1
D(ω ) = δ (ω ) − i
πω
(3.2)
10
Transfer and time domain functions
The proof is straightforward by using Dirac delta’s properties in equation (3.1):
~
1 +∞ δ ( x)
1 +∞ δ ( x)
1
1
1
1
dx = ∫ −
dx = [δ (ω ) * (− )] = [(− ) * δ (ω )]
Im[D(ω )] = ∫
π
ω
π
ω
π −∞ x − ω
π −∞ ω − x
~
1 + ∞ δ (ω − x)
1 +∞ δ ( x − ω )
1
dx = ∫ −
dx = −
⇒ Im[D(ω )] = ∫ −
x
x
π −∞
π −∞
πω
where “*” symbolizes the convolution procedure.
However this intimate relation between the reciprocal function and the delta function
was first noticed by Dirac (1958). He wanted to define more accurately the reciprocal
function in the neighborhood of 0. Thus he imposed an extra condition, limiting in
+ε
that way the value of the integral
1
∫ε x dx =0 by taking into account the anti-
−
symmetry of the reciprocal function.
Figure 3.1 Reciprocal Function
This fundamental relation between the reciprocal and the delta function has a very
important mathematical impact. If the standard expression from differential calculus
is used
d
1
(log x ) = , the result of the integral is − iπ , a result which is
dx
x
contradictory to the aforementioned extra condition of Dirac. Dirac explained that as
x passes through zero this pure imaginary part is vanishing discontinuously. Thus the
11
Transfer and time domain functions
expression given by Dirac (1958) for the derivative of logarithm is
~
d
1
1
(log x) = − iπδ ( x) = −iπ [δ ( x) − i ] = −iπ D( x)
dx
x
πx
(3.3)
3.2 Strictly proper/causal functions
Another very important parameter that is helpful for the characterization of the
functions in the time domain is whether the transfer function is a) strictly proper, b)
simply proper c)improper. In chapter 2 it was mentioned that the constitutive equation
of the linear viscoelastic models is like (2.1):
N
M
N
dm
dn
dm
dn
[ ∑α m m ]P (t ) = [∑ bn n ]u (t ) ⇒ F[[ ∑ α m m ]P (t )] = F[[∑ bn n ]u (t )] ⇒
m=0
dt
n=0
dt
m=0
dt
n =0
dt
M
N
M
N
m=0
n =0
[ ∑α m (iω ) m ]P (ω ) = [∑ bn (iω ) n ]u (ω ) ⇒ P (ω ) =
∑ b (iω )
n =0
M
∑α (iω )
m=0
N
K (ω ) =
∑ b (iω )
n=0
M
u (ω ) ⇒
m
m
n
n
∑α (iω )
m=0
n
n
m
m
where K (ω ) is a generalized transfer function which relates a displacement input to a
force output. If s = iω ⇒ ω =
s
then
i
N
s
K( ) =
i
∑b s
n =0
M
∑α
m =0
n
n
,
m
s
(3.4)
m
s
i
where K ( ) is the ratio of two polynomials. The nominator polynomial is of n
s
i
degree and the denominator one is of m degree. Therefore, K ( ) has n zeros and
m poles [Makris (1996)].
A transfer function that has more poles than zeros ( m > n ) is called “strictly
proper” and results in a “strictly causal” time domain function. A strictly proper
transfer function means that the output does not react instantaneously to an input
modulation [Makris (1996)]; this represents the real physical systems. Even for these
12
Transfer and time domain functions
functions extra care is needed if they have a pole at zero because -as it was analyzed
in the previous section (3.1) - there might be needed a delta function as an additional
term.
Furthermore except from the strictly proper functions when m = n the
transfer function is called simply proper. With these kinds of transfer functions the
first relation of (3.1) does not hold, because it maintains a finite value as frequency
tends to infinity. Also it leads to a time domain function with a singularity at the time
origin; that is, the model instantaneously produces an output at a given input [Makris
(1996)].
The last situation ( m < n ) is when the transfer function is called improper as
it leads in a time domain function that has totally mathematical terms like the
derivative of a singularity in the time origin which has no significant realistic impact.
13
Revision of the parameter solid model
4. REVISION OF THE THREE PARAMETER SOLID MODEL
(POYNTING –THOMSON’S MODEL)
The physical system that Poynting-Thomson’s (PT) model represents is shown
in figure 4.1.
Figure 4.1. The Poynting-Thomson’s model
4.1 Dynamic stiffness and memory function
Starting from the constitutive equation of the model and applying the Fourier
transform
dP(t )
du (t )
dP(t )
du (t )
= k[u (t ) + λ2
] ⇒ F[ P (t ) + λ1
] = F[ku (t ) + kλ2
]⇒
dt
dt
dt
dt
1 + iωλ2
P (ω ) + iωλ1P (ω ) = ku (ω ) + kλ2 iωu (ω ) ⇒ P (ω ) = k
u (ω )
1 + iωλ1
P(t ) + λ1
14
Revision of the parameter solid model
thus the dynamic stiffness of PT’s model is
λ λ ω2
(λ − λ )ω
1
2 = k[
2 ] (4.1)
K (ω ) = k
+ 1 2
−i 1
2
2
2
1 + iωλ
1+ ω λ 2 1+ ω λ 2
1+ ω λ 2
1
1
1
1
1 + iωλ
with m = n , so the dynamic stiffness transfer function is simply proper and the
memory function is,
−
t
λ k
λ
2
2
q (t ) =
δ (t ) + (1 − ) e 1 H (t ) .
λ λ
λ
1
1 1
kλ
(4.2)
4.2 Dynamic flexibility and impulse response function
The dynamic flexibility and its corresponding time domain function is the
impulse response function.
1 + iωλ
1
1
H (ω ) =
=
K (ω ) k1 + iωλ
2
thus the dynamic flexibility is:
(λ − λ )ω
λ λ ω2
1
1
1
2
2 ]
H (ω ) = [
+
+i 1
k 1 + λ 2ω 2 1 + λ 2ω 2
1 + λ 2ω 2
2
2
2
(4.3)
The dynamic flexibility is a simply proper function since the poles are equal with the
zeros. The impulse response function is the inverse Fourier transform of the dynamic
flexibility
−
h(t ) =
t
λ e
1 λ2
δ (t ) + (1 − 1 )
H (t )
λ2 kλ2
k λ1
λ2
(4.4)
The dynamic flexibility is a simply proper function so the impulse response function
is, as expected, almost causal.
15
Revision of the parameter solid model
4.3 Impedance and relaxation modulus
The impedance transfer function is:
Z (ω ) =
1 + iωλ2
K (ω )
=k
iω
iω (1 + iωλ1 )
It can be seen that m > n so the impedance function is strictly proper, thus the
relaxation modulus should be strictly causal.
More detailed,
λ −λ
λλω
1
Z (ω ) = k[ 2 2 1 2 − i (
+ i 1 22 2 )] .
2
2
1 + λ1 ω
ω (1 + λ1 ω ) 1 + λ1 ω
(4.5)
It can be observed that although the impedance function is strictly causal, there is a
pole in zero in the imaginary part that needs caution. So, using the Partial Fraction
Expansion method it can be rewritten in a more appropriate form:
λ1ω
1
1
] − ik
Z (ω ) = k (λ2 − λ1 )[
−i
2
2
2
2
ω
1 + λ1 ω
1 + λ1 ω
(4.6)
Hilbert pairs
Equation (4.6) is in a more appropriate form for deducing that in the real part is
missing the Hilbert transform of the k
1
ω
so that the new correct impedance function
–that its real and the imaginary part are Hilbert pairs- is
Z (ω ) = kλ2 (1 −
λ1
λ1ω
1
1
)[
−i
] + πk[δ (ω ) − i ]
2
2
2
2
λ2 1 + λ1 ω
πω
1 + λ1 ω
(4.7)
The form of (4.7) that is more helpful for computing the inverse Fourier transform
and finding the relaxation modulus.
−
λ
1 1
k (t ) = k[ + sgn(t )] + k ( 2 − 1)e λ H (t )]
2 2
λ1
t
1
(4.8)
The relaxation modulus is strictly causal as expected.
4.4 Mobility and step response function
The mobility function of the PT model is
16
Revision of the parameter solid model
− ω 2 λ1 + iω
1
M (ω ) =
=
Z (ω ) k (1 + iωλ2 )
And more detailed the mobility function is:
λ1λ2ω 3
1 (λ2 − λ1 )ω 2
ω
M (ω ) = [
+ i(
−
)] .
2
2
2
k 1 + λ2 ω 2
1 + λ2 ω 2 1 + λ2 ω 2
(4.9)
This is an improper function. Its inverse Fourier transform is the step response
function m(t )
−
m(t ) =
t
λ
λ
1
e
d δ (t )
[(1 − 1 )δ (t ) + ( 1 − 1)
]
H (t ) + λ1
kλ2
dt
λ2
λ2
λ2
λ2
(4.10)
It is observed that step response function is also almost causal but has terms like the
derivative of the delta function that they do not have any practical effect.
17
Revision of the parameter solid model
4.5 Plots of the transfer and time domain functions
Three Parametric Solid (Poynting-Thomson)
Constitutive Equation
Dynamic Stiffness, K(ω)
Dynamic Flexibility, H(ω)
Impedance, Z(ω)
Mobility, M(ω)
Memory Function, q(t)
Impulse Response Function, h(t)
P(t ) + λ1
dP(t )
dt
K[πδ (ω) +
λ 2ω
λλ ω
1 −i ( 1 +
1
− 1 2 )]
2
2
2
ω 1 + ω 2λ
1 + ω 2λ
1 + ω 2λ
1
1
1
λ −λ
2
2
ω3λ λ
1 ω (λ2 − λ1)
ω
1 2 )]
[
+ i(
+
2
2
2
2
K 1 + ω 2λ 2
1+ω λ
1+ ω λ
2
2
2
λ
−
λ
K
[ λ δ ( t ) + (1 − 2 ) e
2
λ
1
1
1
Kλ
t
λ
1 H ( t )]
−
λ
[ λ δ ( t ) + (1 − 1 ) e
1
λ
2
2
t
λ
2 H ( t )]
t
λ
λ
1 1
K[ + sgn(t) + ( 2 − 1)e 1 H (t )]
λ
2 2
1
λ
Step Response Function, m(t)
]
dt
ω (λ − λ )
ω 2λ λ
1
1 2 −i
1
2 ]
K[
+
2
2
2
2
2
2
1+ω λ
1+ω λ
1+ω λ
1
1
1
λ λ ω2
ω(λ − λ )
1
1
1
2 ]
[
+ 1 2
+i
2
2
K 1 + ω 2λ 2 1 + ω 2λ 2
1+ ω λ
2
2
2
−
Relaxation Modulus, k(t)
d u(t )
= K [u(t ) + λ2
λ
1
1
[(1 − 1 )δ (t ) +
( 1 − 1) e
Kλ
λ
λ λ
2
2
2 2
−
t
λ
2 H ( t ) + λ dδ ( t ) ]
1 dt
Table 4.1. Transfer and time domain functions for Poynting-Thomson’s model
18
Revision of the parameter solid model
Figure 4.2. Plots of the time domain functions of the Poynting-Thomson’s’ model
Figure 4.3. Plots of the frequency domain functions of the Poynting-Thomson’s model
19
Revision of the parameter solid model
4.6 Several comments on Poynting-Thomson’s model
The frequency domain functions are symmetric with respect to zero frequency.
Also it can be deduced that there is symmetry in the force response and in the
velocity response because of the analogous dynamic stiffness and mobility
transfer functions. That symmetry according to Giesekus (1965) is a result of
the fundamental relationship of the dissipative energy expressed as
t
•
Ediss = ∫ P(τ ) u (τ )dτ .
−∞
All the time domain functions are causal (or almost causal).
The aforementioned symmetry is valid, between the force and the velocity
output in the time domain.
The displacement and the velocity output have an instantaneous finite
response to a force impulsive input. Furthermore the aforementioned result
accounts also for the force output resulting from an impulsive displacement
input. All the above, issue from the spring which is connected in series with
the Kelvin-Voigt element.
The relaxation modulus is probably one of the most critical time domain
functions which characterize the behavior of the system. In the specific case
the relaxation modulus (the force) responds instantaneously to a step input
displacement but with a finite value in the time origin corresponding to the
equivalent stiffness of the system. The spring undertakes the total
displacement so the system stabilizes to a finite force output value. It is clear
that the dominant element of the system is the spring (solidlike model).
20
Revision of the three parameter fluid model
5. REVISION OF THE THREE PARAMETER FLUID MODEL
(JEFFREYS’ MODEL)
The physical system that is represented by Jeffreys’ model is shown in figure
5.1.
Figure 5.1. The Jeffreys’ model
4.1 Proof of the constitutive equation
The dashpot c1 and the spring k in the Kelvin-Voigt model have common
displacements but they “share” the force P (t ) . Also the dashpot c 2 and the KelvinVoigt element have the same force F and they “share” the displacement of the
system. Translating the second sentence into equations:
•
P (t ) = c2 u 2 ( t ) ,
•
(5.1)
•
P (t ) = k [u(t ) − u2 (t )] + c1 [u(t ) − u 2 (t )]
•
(5.2)
••
Solving (3.5) for u 2 ( t ) and u 2 ( t ) , differentiating (3.6):
•
u 2 (t ) =
P (t )
,
c2
(5.3a)
21
Revision of the three parameter fluid model
•
••
u 2 (t ) =
•
•
P (t )
,
c2
•
(5.3b)
••
••
P (t ) = k [u(t ) − u 2 (t )] + c1 [u(t ) − u 2 (t )]
(5.4)
Substituting (5.3) into (5.4):
•
•
•
••
P (t )
P (t )
P (t ) = k u (t ) − k
+ c1 u(t ) − c1
⇒
c2
c2
c2 + c1 dP (t )
du (t ) c1 d 2 u (t )
P (t ) +
= c2 [
+
]⇒
k
dt
dt
k dt 2
dP (t )
du (t )
d 2 u (t )
P (t ) + λ1
= η[
+ λ2
]
dt
dt
dt 2
where
λ1 =
(5.5)
c2 + c1
c
is called relaxation time, λ2 = 1 retardation time and η = c 2 is
k
k
called fluidity of the model.
4.2 Dynamic stiffness and memory function
Starting from equation (5.5) and applying the Fourier transform
dP (t )
du (t )
d 2 u (t )
dP (t )
du (t )
d 2 u (t )
P(t ) + λ1
= η[
+ λ2
] ⇒ F[ P (t ) + λ1
] = F[η
+ ηλ2
]
dt
dt
dt
dt
dt 2
dt 2
iωη − ω 2ηλ2
⇒ P (ω ) + iωλ1P (ω ) = ηiωu (ω ) − ηλ2ω 2 u (ω ) ⇒ P (ω ) =
u (ω )
1 + iωλ1
thus the dynamic stiffness of Jeffreys’ model is
iωη − ω 2ηλ2
(λ1 − λ2 )ω 2
λ1λ2ω 3
ω
K (ω ) =
= η[
+ i(
+
)] (5.6)
2
2
2
1 + iωλ1
1 + λ1 ω 2
1 + λ1 ω 2 1 + λ1 ω 2
with m < n , so the dynamic stiffness transfer function is improper and it is expected,
as stated in (3.2), that the time domain function –memory function- has unrealistic
parts like the derivative of a singularity. So the memory function is,
−
q(t ) =
t
η
λ
λ
e
dδ (t )
[(1 − 2 )δ (t ) + ( 2 − 1)
H ( t ) + λ2
]
dt
λ1
λ1
λ1
λ1
λ1
(5.7)
where H (t ) is the Heaviside function.
22
Revision of the three parameter fluid model
4.3 Dynamic flexibility and impulse response function
The inverse of the dynamic stiffness in the frequency domain is the dynamic
flexibility that its corresponding time domain function is the impulse response
function.
u (t ) = h(t ) * P(t ) ⇒ u (ω ) = H (ω )P (ω ) ⇒
1 + iωλ1
1
H (ω ) =
=
K (ω ) η (iω − λ2ω 2 )
thus the dynamic flexibility is:
H (ω ) =
λ −λ
λ1λ 2ω
1
[ 1 2 2 2 − i(
)]
+
2
2
η 1 + λ2 ω
ω (1 + λ 2 ω 2 ) 1 + λ 2 ω 2
1
(5.8)
The dynamic flexibility is a strictly proper function since the poles are more than
zeros, but except the imaginary poles there is a pole at the frequency origin in the
imaginary part of the function. In this case, as it was aforementioned, it needs special
care because the real and imaginary parts are not Hilbert pairs. Thus the imaginary
part needs a special treatment by using the Partial Fraction Expansion of the first term
of the imaginary part of the function:
λ1λ2ω
],
η ω (1 + λ2 2ω 2 ) 1 + λ2 2ω 2
1
Im[H (ω )] = − [
1
+
(5.9)
and the first term (Im1) needs to be decoupled
Im1 = −
−
1
x
y
1
yω
=
+
⇒−
= x+
⇒
2
2
2
2
2
2
2
ηω (1 + λ2 ω ) ηω η (1 + λ2 ω )
1 + λ2 ω
1 + λ2 ω 2
1
1 + λ2 ω
2
= x+
2
ω =0
yω
1 + λ2 ω 2
and for finding y : Im 1 = −
⇒ x = −1
2
ω =0
1
1
y
2
=
+
⇒ y = λ2 ω .
2
2
2
2
ηω (1 + λ2 ω ) ηω η (1 + λ2 ω )
The total imaginary part is
23
Revision of the three parameter fluid model
Im[H (ω )] = −
1
ηω
+
λ2 (λ2 − λ1 )ω
2
η (1 + λ2 ω 2 )
(5.10)
The Hilbert transform of the imaginary part (5.10) is the new real part
Re[H (ω )] = −
λ1 − λ2
1 +∞ Im[H ( x)]
π
dx = δ (ω ) +
2
∫
η
π −∞ x − ω
η (1 + λ2 ω 2 )
(5.11)
As it was expected the pole in the frequency origin in the imaginary part leaded to a
singularity at the origin in the real part. From equation (5.10), (5.11) the total dynamic
flexibility is:
λ −λ
1
1 ( λ − λ 2 ) λ 2ω
H (ω ) = [πδ (ω ) + 1 2 2 2 − i ( + 1
)]
2
η
ω
1 + λ2 ω
1 + λ2 ω 2
(5.12)
And the impulse response function is the inverse Fourier transform of the dynamic
flexibility
−
1 1 1
λ
h(t ) = [ + sgn(t ) + ( 1 − 1)e λ H (t )]
η 2 2
λ2
t
2
(5.13)
where sgn(t ) is the signum function.
The dynamic flexibility is a strictly proper function so the impulse response function
is, as expected, strictly causal.
4.4 Impedance and relaxation modulus
The relaxation modulus filters the input signal of the velocity and gives a force
output.
•
P(t ) = k (t ) * u (t ) ⇒ P (ω ) = Z (ω ) u (ω ) ⇒
Z (ω ) =
K (ω ) η (iω − λ2ω 2 )
=
iω
iω (1 + iωλ1 )
It can be seen that the poles are equal with the zeros m = n and the impedance
function is simply proper, thus the relaxation modulus should be almost causal.
The impedance function more detailed
λ1λ2ω 2
(λ2 − λ1 )ω
Z (ω ) = η[
+
+
i
]
2
2
2
1 + λ1 ω 2 1 + λ1 ω 2
1 + λ1 ω 2
1
(5.14)
24
Revision of the three parameter fluid model
Because it is simply proper function, the value of impedance when frequencies
approaching infinity is finite equal withη .
η (iω − λ2ω 2 )
= η with the help of the “Del’Hospital” theorem. So
lim
Z (ω ) = lim
ω →∞
ω →∞
iω (1 + iωλ1 )
impedance can be rewritten as
Z (ω ) = η − η
(λ1 − λ2 )ω
i (1 + iωλ1 )
(5.15)
The form of (5.15) that is more helpful for computing the inverse Fourier transform
and finding the relaxation modulus.
λ −
η
k (t ) = [λ2δ (t ) + (1 − 2 )e λ H (t )]
λ1
λ1
t
1
(5.16)
It is almost causal as expected and not strictly causal because of its instantaneous
response (delta function) to the input.
4.5 Mobility and step response function
Mobility function is the reciprocal of the impedance in the frequency domain,
thus relating a force input to a velocity output.
•
u (t ) = m(t ) * P (t ) ⇒ u (ω ) = M (ω )P (ω ) ⇒
iω (1 + iωλ1 )
1
M (ω ) =
=
Z (ω ) η (iω − λ2ω 2 )
And more detailed the mobility function is:
(λ − λ )ω
λ1λ2ω 2
M (ω ) = [
+
+ i 1 22 2 ] .
2
2
2
2
η 1 + λ2 ω 1 + λ2 ω
1 + λ2 ω
1
1
(5.17)
It is a simply proper function like impedance as they are reciprocal to each other. Its
inverse Fourier transform is the step response function m(t )
λ −
m(t ) =
[λ1δ (t ) + (1 − 1 )e λ H (t )]
λ2
ηλ 2
1
t
2
(5.18)
It is observed that step response function is also almost causal because of its
instantaneous response to the input.
25
Revision of the three parameter fluid model
5.6 Plots of the transfer and time domain functions
Three Parametric Fluid (Jeffreys’ model)
Constitutive Equation
Dynamic Stiffness, K(ω)
Dynamic Flexibility, H(ω)
Impedance, Z(ω)
Mobility, M(ω)
Memory Function, q(t)
2
d u(t )
P(t ) + λ1
= η[
+ λ2
]
2
dt
dt
dt
dP(t )
ω 2 (λ − λ )
ω 3λ λ
ω
1
2
1 2 )]
+i (
+
η[
2
2
2
2
2
2
1+ω λ
1+ ω λ
1+ω λ
1
1
1
1
η
[πδ (ω) +
λλ ω
ωλ 2
2 −i ( 1 + 1 2
2
−
)]
2
2
2
2
2
2
ω
1+ ω λ
1+ ω λ
1+ ω λ
2
2
2
λ −λ
1
λ λ ω2
ω (λ − λ )
1
2
1 ]
+ 1 2
+i
η[
2
2
2
2
2
2
1+ ω λ
1+ω λ
1+ ω λ
1
1
1
ω2λ λ
ω(λ − λ )
1
1 2 +i
1 2 ]
[
+
η 1 + ω2λ 2 1 + ω2λ 2 1 + ω2λ 2
2
2
2
1
t
−
λ
λ
η
1 λ2
dδ ( t )
[(1 − 2 )δ ( t ) +
(
− 1) e 1 H ( t ) + λ
]
2 dt
λ
λ
λ λ
1
Impulse Response Function, h(t)
du(t )
1
1
1
t
−
λ
λ
1 1 1
[ + sgn(t ) + ( 1 − 1)e
λ
2
2
η 2
Relaxation Modulus, k(t)
t
−
λ
λ
η
1 H ( t )]
[ λ δ ( t ) + (1 − 2 ) e
2
λ
λ
1
Step Response Function, m(t)
2 H (t )]
1
ηλ
1
λ
[ λ δ ( t ) + (1 − 1 ) e
1
λ
2
2
−
t
λ
2 H ( t )]
Table 5.1.Transfer and time domain functions for Jeffreys’ model
26
Revision of the three parameter fluid model
Figure 5.2. Plots of the time domain functions of the Jeffreys’ model
Figure 5.3. Plots of the frequency domain functions of the Jeffreys’ model
27
Revision of the three parameter fluid model
5.4 Several comments on Jeffreys’ model
The frequency domain functions are symmetric with respect to zero frequency.
Also it can be deduced that there is symmetry in the force response and in the
velocity response because of the analogous dynamic stiffness and mobility
transfer functions.
All the time domain functions are causal (or almost causal).
The aforementioned symmetry is again valid, between the force and the
velocity output in the time domain.
The force as an output, in both displacement and velocity input cases has an
instantaneous response expressed as a delta function in the time origin. The
same accounts for the velocity as an output to a force input. All the above,
result from the dashpot’s contribution to the total response of the system.
The main difference from Poynting-Thomson’s model –essentially the
difference between a solid and a Newtonian fluid- is the behavior of the
system itself expressed with the relaxation modulus. In the specific case, after
its instantaneous increase, it decreases monotonically. That means that the
force as an output to a velocity impulse input or a displacement step input
(Heaviside function) is increasing infinitely at t=0 because of the dashpot’s
initial “violent” resistance and not with a finite value like in the PoyntingThomson’s model. Furthermore the system relaxes as the velocity input drops
to zero and the dashpot resistance force fades out proportionally, thus resulting
in an imbalanced response since the step displacement is undertaken by the
dashpot (fluidlike model).
28
Admissibility criteria for the relaxation modulus
6. ADMISSIBILITY CRITERIA FOR THE RELAXATION
MODULUS
From theoretical analysis and experimental work it was found that the
relaxation modulus is a very important function, for viscoelastic models, that has to
satisfy – except from the fundamental causality criterion- specific criteria.
Furthermore the relaxation function has been observed to be positive and monotonic
decreasing [Akyildiz et al. (1990)]. Thus
k (t ) ≥ 0,
(6.1)
dk (t )
<0
dt
In this thesis both these additional criteria are satisfied but not for both models
simultaneously, as for satisfying these criteria the relative ratio between relaxation
and retardation time plays a very important role. For example the relaxation function
of Jeffreys’ model is positive
λ −
η
k (t ) = [λ2δ (t ) + (1 − 2 )e λ H (t )] > 0,
λ1
λ1
t
1
when λ1 > λ2 . Furthermore when relaxation time is greater than retardation time
λ1 > λ2 then the relaxation modulus is monotonically decreasing for Jeffreys’ model.
Thus the first derivative of the relaxation modulus is negative for λ1 > λ2 .
λ −
λ −
dk (t ) η
dδ (t ) 1
= [λ 2
− (1 − 2 )e λ H (t ) + (1 − 2 )e λ δ (t )] < 0,
dt
dt
λ1
λ1
λ1
λ1
t
1
t
1
From thermodynamic arguments it can be deduced that the work done in any
strain path is starting from equilibrium is non-negative
t
•
W = ∫ σ (τ ) γ (τ )dτ ≥ 0 .
(6.2)
−∞
The energy condition (6.2) leads to another criterion that has to do with the convexity
of the relaxation modulus. However it was proven that the aforementioned criteria
(6.1), (6.2) are not sufficient to describe a physical system of combined springs and
dashpots [Akyildiz et al. (1990)]. Thus it needs an additional criterion coming from
Bernstein theorem (1928) that results to
29
Admissibility criteria for the relaxation modulus
(−1) n k ( n ) (t ) ≥ 0 , n = 0,1,2,...
(6.3)
so that if (6.3) is satisfied - the relaxation modulus is a completely monotone function
with derivatives that alternate in sign- then the mechanical analog model can be
completely equivalent to the integral representation of linear viscoelasticity [Beris and
Edwards (1993)]. In Jeffreys’ model, for example, the relaxation modulus satisfy (6.3)
because
λ −
1
η
k (t ) = [(−1) n ( ) n (1 − 2 )e λ H (t ) + C (t )],
λ1
λ1
λ1
t
(n)
1
(6.4)
where in C (t ) are all the terms that have the delta function or its derivatives and they
are all in the neighborhood of 0 thus they do not play any crucial role in the (6.3)
criterion. Furthermore equation (6.4) satisfies the above criterion for λ1 > λ2 , so
Jeffreys’ model relaxation modulus is applicable in linear viscoelastic theory with
relaxation time larger than the retardation time.
As far as it concerns the Poynting-Thomson model, the exact opposite
situation is valid. That is, that although the first criterion is satisfied, the second
criterion of equation (6.1) is not satisfied for λ1 > λ2 as the relaxation modulus is
monotonically increasing. Also the (6.3) criterion is satisfied for retardation time
greater than relaxation time λ2 > λ1 .
So it is obvious after the above thoughts that both Jeffreys’ and PoyntingThomson’s models cannot be used for modeling the same mechanical system as they
do not meet the same admissibility criteria for standard system parameter values.
30
Conclusions
6. CONCLUSIONS
Two widely used viscoelastic models have been investigated in depth in this
thesis; the three parameter fluid and solid model -Jeffreys’ and Poynting-Thomson’s
model- for which all transfer and basic time response function have been examined.
The first step was to derive the correct transfer function of its model in the
frequency domain, as it was assumed that the deformations/strains were appropriate
for using linear properties of the Fourier transform. The transfer functions were
complex thus their real and imaginary parts had to satisfy the Kramers-Kronig
relationships for fulfilling the causality criterion according to the fundamental relation
between the analyticity and the causality. However the problem was more specific. In
two situations the imaginary part of the transfer function had a hidden pole in the
frequency origin that was revealed with the use of partial fraction expansion method.
The result was an additional hidden Dirac delta function in the real part, as the delta
function is the Hilbert pair of the pole in zero.
Consequently, the time domain functions of both models were causal (or
almost causal) satisfying in that way successfully causality requirement. From
examining the time domain functions and especially the relaxation modulus one can
find the dominant element of the model (spring, dashpot, etc) thus understanding its
basic response to every kind of excitation. However, except the aforementioned
criterion, additional requirements (that have to do with the relaxation modulus) have
been investigated for the admissibility of the models. It is clear that for different
relative values of the parameters of the models like relaxation and retardation time,
different model has to be valid. They cannot be reliable for the same values of those
parameters as they represent physically different conditions.
Finally, a more solid establishment of the linear viscoelastic theory has been
done and applied in the three parametric fluid and solid as a continuation of Makris
(1996), thus making them fully available for applications in disciples like engineering
seismology and structural dynamics.
31
References
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•
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•
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33