-
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
FITTING AND INTERCONVERTING PRONY SERIES OF
VISCOELASTIC ENGINEERING MATERIALS
USING A COMPUTER PROGRAM
Henrique Nogueira Silva
Jorge Barbosa Soares
[email protected]
[email protected]
Centro de Tecnologia - Universidade Federal do Cear
Bloco 703 - 60455-760, Fortaleza, Cear, Brazil
Flvio Vasconcelos de Souza
[email protected]
PRH-31/ANP, Universidade Federal do Cear
Bloco 717 - 60455-760, Fortaleza, Cear, Brazil
Flvio Mamede Pereira Gomes
[email protected]
Eletrobras/Furnas. Laboratrio de Concreto de Goinia
Rodovia BR-153, S/N - Zona Rural, Goinia, Gois, Brazil
Abstract. Numerical and experimental results may differ
considerably if it is considered sim-
ple constitutive models, such as elastic or elastoplastic. For a
certain class of materials, a
viscoelastic behavior results in a closer approximation between
simulation and experiments
over time. Despite a more realistic representation, the theory
of viscoelasticity requires more
effort with respect to numerical manipulation of the input data.
Curve fitting of Prony (Di-
richlet) series of viscoelastic properties and interconverting
between them is a nontrivial task,
with numerical instability problems, and prone to error in a
spreadsheet due to manipulation
of many coefficients. Using ViscoLab, a computer program that
allows faster fitting and in-
terconverting viscoelastic properties using Prony series, it is
shown two practical applica-
tions of curve fitting and interconversion: (i) characterization
in time domain of Creep Com-
pliance and Relaxation Modulus of an early age mass concrete
used in dam construction; and
(ii) constitutive characterization in frequency domain, from
Dynamic Modulus to Relaxation
Modulus, for an asphalt concrete mixture used in pavement
surface courses. The study re-
ports all necessary steps to adequately characterize
viscoelastic engineering materials, con-
tributing to widespread the applicability of the theory of
viscoelasticity in numerical modeling
of civil engineering problems.
Keywords: Viscoelasticity, Prony series, Early age concrete,
Asphalt concrete
mailto:[email protected]:[email protected]:[email protected]:[email protected]
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Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
1 VISCOELASTIC CONSTITUTIVE MODELLING
To materials that follows the theory of viscoelasticity, stress
and strains are computed
based on all loading history and, in simplified notation of
one-dimensional case, these quanti-
ties are computed using convolution integral according to Eq.
(1)(Ferry, 1980; Schapery,
1982):
( ) ( )
( ) ( )
(1)
Where ( ) and ( ) are instantaneous stress and strain,
respectively; is time like vari-able used in integration; and ( )
and ( ) are Relaxation Modulus and Creep Compliance, respectively,
elementary viscoelastic properties in time domain, which are
efficiently repre-
sented by Prony series (or Dirichlet series) according to Eq.
(2) (Zocher, 1995), where and are independent terms; and are
dependent terms; and and time constants.
( )
( ) ( )
(2)
The need to define N exponential terms in equation above is
based on experimental ob-
servations that viscoelastic behavior is developed in different
logarithmic time scales, so each
term represent the behavior in its specific time scale. Even
though computationally efficient,
fitting Prony series generally results in numerical
instabilities with some negative coefficients
or (Kim, 2008) or even a totally disagreement of fitted curve to
experimental data (Sil-
va, 2009). A robust manner to fit Prony series is performing
restricted nonlinear least squares,
considering classical or evolutionary algorithms, including all
terms , and , = 1 to N (or , and , = 1 to N) restricted to be all
positive values. An evenly effective and more
simplified technique is setting independent term (or ) and time
constants (or ), then
Eq. (2) is linearized, and then a linear least squares can be
performed using Eqs. (3) and (4).
( ( ) )
( )
(3)
( ) (
)
( ( ) ) ( )
( )
(4)
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Where ( ) and ( ) are experimental data in each observed time (
= 1 to M). For Eqs.
(3) and (4) the number of experimental data must be greater than
the number of Prony terms
(i.e. M > N), and this is the typical situation found in
practice. For case of M = N, cited nu-
merically instabilities are accentuated and its equations will
not be shown in this paper for
brevity, but can be seen in Schapery (1982) and Silva
(2009).
Other task commonly performed in constitutive modeling of
viscoelastic materials is the
interconvertion between elementary properties in time and
frequency domain motivated by
mechanical and/or time limitations in experimental programs
(Ferry, 1980; Kim, 2008; Park
& Schapery, 1999; Silva, 2009). In practice, the Relaxation
Modulus indicated in first part of
Eq. (2) is a difficult test to be performed once a controlled
stress level is not well established
in servo-hydraulic machines (Kim, 2008). This way, ( ) is
indirectly obtained by intercon-
verting from ( ), an easier strain controlled test. Other
example is performing accelerated
frequency domain tests indicated by Eqs. (8) and (9) as follows,
to obtain Relaxation Modulus
( ), the property asked in Finite Element Programs to execute
numerical analysis (Kim,
2008; Park & Schapery, 1999; Silva, 2009).
For interconverting from Relaxation Modulus ( ) to Creep
Compliance ( ), both rep-
resented by Eq. (2), Park & Schapery (1999) and Silva (2009)
performed algebraic manipula-
tions and found the Eq. (5), a linear system to be solved by
pre-selecting time constants of
target function ( ) ( in this case). In this most simplified
case the sampling points are se-
lected using = , where typically = 1 or = 1/2 is used.
( )
where,
( )
( )
(
)
and
[
] [
]
(5)
For the inverse case, and most practical in an experimental
perspective, interconverting
from Creep Compliance ( ) to Relaxation Modulus ( ) is given by
Eq. (6). Similarly, it must be pre-selected time constants of
target function E(t) and sampling points by a crite-ria, like = ,
where typically = 1 or = 1/2 is used.
( )
where, (6)
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Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
( ) (
)
( ) (
)
(
)
and
[ ( )
]
For terms and where in Eqs. (5) and (6), there is a numerical
way to find-
ing time constants of target function (or ) if source Prony
series is available , and
(or , and ). It is based on Carson transform of Prony series in
Eq. (2) that results in Eq.
(7), where is the Laplace space.
( )
( )
(7)
This way, Park & Schapery (1999) show that plotting in a
log-log scale the absolute val-
ues of Eq. (7) (i.e. | ( )| or | ( )|) versus the inverse
reciprocal of Laplace space ( ,
with assuming negative values) it results in a graph with
maximum values whose abscissa is and minimum values whose abscissa
is . Extensive use of this technique can be found in
Silva (2009).
For viscoelastic properties in frequency domain, the real and
imaginary parts of Complex
Modulus and Complex Compliance are given by Eqs. (8) and (9),
respectively, where is he frequency in Hz. Theses equations were
demonstrated in a classical papers using Fourier
transforms (Park & Schapery, 1999; Schapery, 1982). Once the
time constants ( or ) of
target function ( ( ) or ( )) are appropriately chosen, a linear
system can be solved using Eqs. (8) or (9) to obtain Prony
coefficients ( or ). This way, fitting these equations it is,
in
fact, performing interconvertion from frequency domain to time
domain . If using Eq. (9), it is important to cite that two steps
of interconvertion must be used to obtain Relaxation
Modulus, from ( ) or ( ) to ( ) and finally to ( ). According to
Kim (2008) the real part ( ) or ( ) are preferable to obtain more
consistent results in interconverting to find Prony series in Eq.
(2).
( )
( )
(8)
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
( )
( )
(9)
For polymeric materials, such as asphalt concrete and asphalt
binder, it is common use
Time-Temperature Superposition Principle (TTSP) based on the
strong dependence of its vis-
coelastic behavior in relation to temperature (Brinson &
Brinson, 2010; Ferry, 1980; Lytton et
al., 1993). This principle states that an increase in
temperature is nearly equivalent to an in-
crease in time. In this manner, one can perform experiments in
short time and various temper-
atures and reduces the viscoelastic characterization to a single
(chosen) temperature in a wide
range in time, named reduced time = / , where is the horizontal
shift factor calculat-
ed from Arrhenius equation expressed in Eq. (10).
( ) (
) (10)
Where is the reference temperature; and is Arrhenius constant,
with specific values for each type of material.
For Portland concrete, the viscoelastic behavior is noticeable
in early ages, so this behav-
ior must be taken in account when modeling, for example, phase
construction of dams or
large foundations blocks. As the age of Portland concrete
advances, experimental results indi-
cate the Prony coefficients and must not be considered constant
anymore and Eq. (2) is
modified in order to modeling evolution of Relaxation (or Creep)
according to the age of con-
crete , resulting in general Eq. (11).
( ) ( ) ( )
( ) ( ) ( ) ( )
(11)
Bofang (2014) shows a possibility to modeling evolution with age
of independent term ( ) and dependent terms ( ) according to Eq.
(12). It is important to cite that the third
term ( ) can be omitted according to experimental data. Similar
equation can be expressed to Relaxation Modulus terms ( ) and ( ) (
= 1 to 3).
( )
( ( )) ( )
( )
and
( )
( )
(12)
2 FITTING AND INTERCONVERTING USING VISCOLAB
In order to enable a faster process of fitting and
interconverting Prony series according to
Eqs. (3), (4), (5), (6), (8) and (9) a software was used in the
present paper, ViscoLab (Multi-
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
mechanics, 2011). The program is available in Windows and Linux,
and can be easily ported
to Mac OS, once it was developed using Qt framework (Qt Project,
2014). As can be seen in
Figure 1, in the main window the user must select the
viscoelastic property in section Data
Type, and then import experimental data from a spreadsheet,
whose values are shown in sec-
tion Sample and can be graphically represented in Figure 2.
Figure 1. Main window of ViscoLab in Linux Ubuntu
Figure 2. Plot window of ViscoLab in Linux Ubuntu
This graphical representation enables the user to select the
number of Prony terms to
reach all range of experimental data, as well as to graphically
identify the value for the inde-
pendent term of the Prony series (i.e. or ). In the section
Collocation the user must enter values for time constants of Prony
series ( or ) and, after pressing a button, the
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Prony coefficients (i.e. or ) are estimated by Eqs. (3) and (4).
As can be seen in Figure 1
the user can also enter values for time constants ( or ) for the
Interconvertion section
and easily estimate Prony coefficients of the target
interconverted viscoelastic property ac-
cording the Eqs. (5) and (6). Figure 1 shows a benchmark of M =
52 data points of source
data for a typical relaxation modulus of a specific polymer,
according to (Park & Schapery,
1999). It can be seen in Figure 2 a good agreement in fitted
Prony series of Relaxation Modu-
lus and the interconverted property Creep Compliance using N =
11 Prony terms. For other
benchmarks the reader is referred to (Multimechanics, 2011),
which shows fitted and inter-
converted functions in time domain by Eqs. (5) and (6)
(Relaxation Modulus and Creep Com-
pliance) and in frequency domain by Eqs. (8) and (9) (real and
imaginary parts of Complex
Modulus and Complex Compliance).
3 FITTING AND INTERCONVERTING PRONY SERIES OF EARLY AGE MASS
CONCRETE
In order to illustrate the process of fitting a Prony series of
Creep Compliance ( ) of early age Portland cement concrete it was
used experimental data available in Bofang (2014),
as shown in Figure 3. The figure shows experimental data of
Creep Compliance ( ) = ( ) of mass concrete used in construction of
a Chinese dam, Felly Cangon, as a function of the
age of the concrete (time from 0 to 560 days) and age of loading
(time for 2, 7, 28, 90 and 365 days). A similar picture can be
found in Bazant (1988).
Figure 3. Experimental data of Creep Compliance of mass concrete
of a Chinese dam
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Figure 4. Experimental data of Creep Compliance of mass concrete
of a Chinese dam
numerically mapped using PlotDigitizer
Finding numerical values of experimental data of viscoelastic
functions of early age con-
crete is a hard task. Papers usually show these properties only
in a graphical manner, not in-
cluding numerical values. To overcome this difficulty, the data
points in Figure 3 were
mapped using PlotDigitizer, which is a software used to digitize
scanned plots of functional
data and allows to take a scanned image of a plot (in GIF, JPEG,
or PNG format) and quickly
digitize values of the plot simply by clicking the mouse on each
data point (Plot Digitizer,
2014). The numbers can then be saved to a text file and used for
numerical manipulations.
The numerical values resulted from mapping Figure 3 are shown at
the end of this paper, in
Appendix and graphically represented in Figure 4. By comparing
Figure 3 and Figure 4 it is
evident the high accuracy of this mapping technique, which
enables one to show details in
fitting Prony series in aging materials.
The first step in fitting Prony series is defining the number of
coefficients ( and ) to
characterize all experimental data range. As the age of concrete
varies from 0 to 560 days in
Figure 4, then it can be chosen 3 coefficients for to represent
all spectrum of time range. In
fact for independent term, for the first age decade around 10
days, and for the
second age decade around 10 days. For further details of number
of coefficients choice the
reader is referred to Silva (2009).
The second step is defining the value for the independent term
for each age of loading. According to Figure 5, a graph with linear
scale of Creep Compliance and log scale of auxil-
iar time , values of can be inferred by power regression. It
seems a good choice considering = 0.1 day for each age (of
loading), then values for are 45.5, 28.9, 28.7, 22.6 and 20 MPa,
for ages (of loading) 2, 7, 28, 90 and 365 days, respec-
tively.
-
Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Figure 5. Defining by power regression
When plotting selected values of for each age of loading, it can
be observed a non-
smooth curve (blue curve in Figure 6), affected mainly due to
the outlier in for 7 days of
age of loading. This non smoothness in over age of loading is
propagated to values ob-
tained to and using Prony series curve fitting with ViscoLab, as
indicated in the yellow
area of green and red curves in Figure 6. The time coefficients
used were = 5 days and =
200 days.
In order to get smoother curves of ( = 1, 2) the simpler
technique is defining visually
new values for . This can be observed in Figure 7, where
smoother selected values 45.5,
35.0, 27.0, 22.6 and 20.0 MPa resulted in smoother curves for (
= 1, 2), with high coeffi-
cient of determination R in ( ) prediction (above 97%). This
visual adjustment of is
acceptable due to three fonts of errors in initial values for
Creep Compliance: (i) mapping
technique is more prone to error in short time; (ii) ramp effect
of real loading of applied stress
could not be taken in account as explained in Silva (2009, p.
23); (iii) Creep Compliance for
initial age of loading (i.e. 2 and 7 days) is strongly affected
by environmental conditions, so
small changes in temperature and/or humidity can significantly
change initial values of Creep
Compliance, and consequently.
According to Figure 7, the Prony series of Creep Compliance of
aging mass concrete can
be represented by Eq. (13), in accordance to general Eq. (12)
suggested by Bofang (2014):
( )
( ( ) )
( ( )
) ( )
(13)
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Where is the age of concrete and is the age of loading, both
expressed in days. It was obtained a small improvement in including
an independent term in ( ) ( = 0, 1, 2), and the final expression
of Creep Compliance of early age mass concrete is described by Eq.
(14),
with and expressed in days:
( ) [ ] [ ] ( ( ) )
[ ] ( ( )
) ( ) (14)
Figure 6. Prony coefficients of Creep Compliance in different
ages of loading
non-smooth curves
Figure 7. Prony coefficients of creep compliance in different
ages
smoother curves
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Equation (14) above plotted against experimental data is shown
in Figure 8. It can be ob-
served a good agreement in overall time of all ages (of loading)
available.
Figure 8. Creep Compliance fitted by Prony series with various
age of loading
As mentioned in section 1 the final output in viscoelastic
constitutive characterization to
perform mechanistic Finite Element Analysis is the Relaxation
Modulus represented by Prony
series. Using ViscoLab, it was interconverted fitted Prony
series of Creep Compliance (in
Figure 7) to Relaxation Modulus, also represented by its
appropriated Prony series. By con-
sidering the simplified choice in time constants of Relaxation
Modulus (i.e. ) it is ob-
tained Prony coefficients ( = 1, 2) as functions of age of
loading according to Figure 10. On the other hand, if considering a
more accurate choice in time constants of target function
(i.e. according to Figure 9 based on ( ) described Eq. (7)) it
can be found Prony
coefficients ( = 1, 2) as smoother functions of age of loading
according to Figure 11. For further explanations of time constants
choice of interconverted function the reader is referred
to Appendix B of Park & Schapery (1999) and Silva (2009, p.
73).
Figure 9. More accurate choice <
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Figure 10. Prony coefficients of relaxation modulus in different
ages
non smooth and due to choice =
Figure 11. Prony coefficients of relaxation modulus in different
ages
smoother and curves due to choice <
The numerical values of Prony series coefficients of both source
Creep Compliance and
target Relaxation Modulus are indicated in Table 1. These values
were graphically illustrated
as previously shown in Figure 7 and Figure 11. It can be
observed in Table 1 a small change
in time constants of target function as age of loading varies.
So, without loss of accuracy, it can be considered the mean values
= 3.10 days and = 146 days in an expression to represent the
overall behavior of aging Relaxation Modulus of mass concrete
according to Eq.
(15), where and are expressed in days:
( ) ( )
( ) ( )
(15)
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Table 1. Prony series of Creep Compliance and Relaxation
Modulus
of early age concrete - choice <
Creep Compliance D(t' = t - ) Relaxation Modulus E(t' = t -
)
(days) 2 (days) 2
(days) (10E-6 MPa-1
) (days) (10E6 MPa)
0 - 45.5 0 - 0.00868892
1 5.00 36.4337 1 2.77 0.00986935
2 200 33.1554 2 143 0.00340546
(days) 7 (days) 7
(days) (10E-6 MPa-1
) (days) (10E6 MPa)
0 - 35.0 0 - 0.0111916
1 5.00 25.8497 1 2.86 0.0123130
2 200 28.5030 2 137 0.00507443
(days) 28 (days) 28
(days) (10E-6 MPa-1
) (days) (10E6 MPa)
0 - 27.0 0 - 0.0166910
1 5.00 16.2103 1 3.11 0.0140863
2 200 16.7023 2 145 0.00627551
(days) 90 (days) 90
(days) (10E-6 MPa-1
) (days) (10E6 MPa)
0 - 22.6 0 - 0.0222007
1 5.00 11.4137 1 3.31 0.0149882
2 200 11.0299 2 151 0.00704977
(days) 365 (days) 365
(days) (10E-6 MPa-1
) (days) (10E6 MPa)
0 - 20.0 0 - 0.0265695
1 5.00 8.73783 1 3.47 0.0153671
2 200 8.89927 2 153 0.00806364
It can be obtained substantial improvement in coefficient of
determination R in functions
( ) ( = 0, 1, 2) in Figure 11, if one considers independent
terms. In fact R was greater than
95% in all three curves and resulted in some negatives
coefficients of ( ), without loss of
accuracy though. This way the final expression of Relaxation
Modulus of early age mass con-
crete is describe by Eq. (16), with and expressed in days:
( ) [ ] [ ] ( )
[ ] ( )
( ) (16)
Using Eq. (16) it can be generated predictions of Relaxations
Modulus in any age of load-
ing , not restricted only to experimental data. This way
interpolations and extrapolations
were performed in Figure 12 (linear scale) and Figure 13
(log-log scale). In order to not su-
perestimate Relaxation Modulus in higher age of loadings
(greater than 28 days), it was con-
sidered the initial age for plotting each curve that equivalent
to 10% of increment in age of
loading (e.g. 120 days, initial age for plotting equal to 132
days = 1,10 * 120 days). It makes
sense because in a higher age of loading, there is no need to
investigate extremely short time
steps once the high strength of concrete in that ages imply in a
prominent ramp test in creep
experiments.
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Figure 12. Prediction of Relaxation Modulus for various age of
loadings
linear scale
Figure 13. Prediction of Relaxation Modulus for various age of
loadings
log-log scale
4 FITTING AND INTERCONVERTING PRONY SERIES OF ASPHALT
CONCRETE
For asphalt concrete viscoelastic characterization it was used
experimental data from an
asphalt mixture of North Caroline, USA, provided by Petrobras
Research Center (Cenpes)
which has an ongoing research with NCSU. The Dynamic Modulus | (
)| and phase angle
( ) collected for 3 temperatures (4, 20 and 40 C) are shown in
Figure 14. It is important to
mention that this experiment using frequency domain is faster
than those performed in time
domain, which results in saving time and costs in a large
experimental program. In this figure
it is also shown a master curve of these properties constructed
from Eq. (10) using an Arrhe-
nius constant equal to 13.000 according to literature (Lytton et
al., 1993). The reference tem-
perature was 30C, for demonstration purposes. Based on inferior
values of | | it
can be inferred an asymptotic value of 500 MPa for .
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Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
(a) Dynamic Modulus
(b) Phase angle
Figure 14. Experimental data of Complex Modulus of an asphalt
concrete
in a typical pavement
Comparing Figure 14 and Figure 4 it can be seen that asphalt
concrete has a wider viscoe-
lastic range than early age concrete, mainly due to master curve
based in time-temperature
superposition principle commonly used for asphaltic materials
characterization. This way,
instead of using just 2 or 3 terms in Prony series, for adequate
viscoelastic characterization of
such material it must be used, at least, 8 terms, one term for
each logarithmic frequency. It is
also important cite that [days] is more appropriate unit in
modelling viscoelastic behavior of
Portland concrete and whereas [seconds] is the unit used to
represent asphalt concrete. These
are related to the numerical simulations of specific area, once
loadings in Portland concrete
are developed along days and for asphalt concrete the time scale
of load pulses from vehicles
is expressed in seconds.
Next step is considering time constants for this 8 terms Prony
series to be fitted according
to Eq. (8). Even though mathematically incorrect, it is
practical to consider time reciprocal
frequency (i.e. ) in estimating time range. For experimental
data available, the max-
imum/minimum frequency of master curve is 2.7E5 Hz/4.3E-3 Hz,
resulting in time range
between 3.8E-6 s and 2.4E2 s. The first main column of Table 2
shows the interconverted
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Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Prony series of Relaxation Modulus, considering the most simple
time constant choice,
s ( = 1 to 8).
If one considers the smoothness of versus according to Silva
(2009, p. 132) it can
be found a smoother representation of Relaxation Modulus. Also,
all interconverted properties
are more prone to obtain good results when trying to obtain a
smooth curve of versus
(or versus ). Using this approach, it was performed other two
interconversions as indicat-
ed in second and third main columns of Table 2, with 8 and 9
terms. Note that a small change
in time constants within its logarithmic time decade results in
smoother versus graph
(see blue region of Figure 15). For a Finite Element Analysis of
this asphalt mixture it then
can be used 9 terms Prony series of Relaxation Modulus indicated
in third main column of
Table 2 and in blue curve of Figure 16.
Table 2. Interconverted Prony series of Relaxation Modulus of
asphalt concrete
with different choices of time constants
i i (s) Ei (MPa) i i (s) Ei (MPa)
i i (s) Ei (MPa)
1 1,00E-05 3710
1 1,00E-05 3713
1 1,00E-05 3552
2 1,00E-04 3557
2 1,00E-04 3549
2 9,00E-05 3551
3 1,00E-03 3611
3 1,00E-03 3629
3 9,00E-04 3325
4 1,00E-02 3464
4 1,00E-02 3396
4 6,00E-03 2673
5 1,00E-01 2024
5 1,00E-01 2279
5 3,00E-02 1990
6 1,00E+00 934
6 3,00E+00 1306
6 2,00E-01 1569
7 1,00E+01 1095
7 7,00E+00 421
7 3,00E+00 985
8 1,00E+02 261
8 1,00E+02 363
8 7,00E+00 614
- 500 - 500 9 1,00E+02 352
- 500
Figure 15. Graphical representation of interconverted Prony
series
of Relaxation Modulus of asphalt concrete
-
Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
Figure 16. Real component of Complex Modulus and
interconverted Relaxation Modulus (9 terms)
For this asphalt concrete mixture, considering much more than 9
terms is an effort that
generally does not result in any improvement of interconverted
Prony series. Silva (2009) has
shown that the adjusted coefficient of determination does not
change as the number of
terms increases, and negative values generally appear. As
previously cited in Section 1, these
negative values result in numerical instabilities of
interconversions using Eqs. (5) to (9), and
in most severe cases result in numerical representations of
Prony series in totally disagree-
ment with the experimental data.
5 CONCLUSIONS
In this paper it was shown all necessary steps to perform
adequate viscoelastic constitu-
tive modelling of materials largely employed in civil
engineering assemblies, such as flexible
pavements and mass concrete dams. For mass Portland concrete it
was fitted Prony series of
Creep Compliance and then interconverted to Relaxation Modulus,
considering the aging ef-
fect in Prony coefficients by a two-step fitting process in both
viscoelastic properties. For as-
phalt concrete it was shown with details results of Complex
Modulus in frequency domain
and the interconvertion to obtain Relaxation Modulus of this
material. More related to the
numerical techniques employed, it was demonstrated that small
changes in pre-choosing time
constants of functions in time or frequency domain can result in
significant improvements of
fitted and interconverted coefficients of Prony series.
-
Fitting and interconverting prony series of viscoelastic
engineering materials using a computer program
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
APPENDIX NUMERICAL VALUES OF EXPERI/MENTAL DATA
CREEP COMPLIANCE OF EARLY AGE CONCRETE OF MASS CONCRETE AGE OF
LOADING:
2 DAYS
AGE OF LOADING:
7 DAYS
AGE OF LOADING:
28 DAYS
AGE OF LOADING:
90 DAYS
AGE OF LOADING:
365 DAYS
age of
concrete
(days)
creep com-
pliance J
(10E-6 per
MPa)
age of
concrete
(days)
creep com-
pliance J
(10E-6 per
MPa)
age of
concrete
(days)
creep com-
pliance J
(10E-6 per
MPa)
age of
concrete
(days)
creep com-
pliance J
(10E-6 per
MPa)
age of
concrete
(days)
creep com-
pliance J
(10E-6 per
MPa)
2,480 51,721 7,891 36,803 28,1804 31,721 90,599 25,2459 365,316
21,721
4,058 60,246 8,440 39,426 29,9839 35,656 91,981 29,1803 366,345
24,344
4,734 65,000 9,243 41,230 38,3253 41,475 94,010 30,9016 369,726
26,148
7,214 68,197 9,018 43,197 49,8229 43,115 96,490 31,2295 376,264
26,639
8,792 71,967 13,978 47,787 73,7198 46,148 99,420 32,1311 425,411
31,557
10,145 77,869 14,654 53,853 100,773 47,541 110,467 33,9344
449,082 32,049
13,527 79,262 21,192 57,623 149,469 52,049 119,259 34,5902
474,783 32,705
18,035 81,066 24,122 61,393 174,493 51,803 150,596 36,3934
499,356 33,115
23,672 83,853 29,308 64,262 200,193 54,426 175,395 38,2787
524,605 33,279
34,493 85,574 33,591 66,639 225,443 55,000 199,291 39,0164
550,757 34,016
50,274 90,246 38,776 67,951 248,663 55,082 227,021 39,8361
75,073 94,262 43,961 68,525 274,815 55,328 249,791 40,082
99,420 97,213 49,372 70,164 298,937 55,328 274,138 40,4918
125,121 97,787 74,622 72,049 325,314 55,656 299,388 41,0656
147,891 98,853 99,871 74,508 348,986 56,557
172,689 101,148 124,219 73,771 373,559 56,885
200,193 102,541 147,665 76,475 400,161 56,393
224,541 102,787 175,845 77,049 424,283 56,639
249,791 103,770 199,066 77,295 446,151 57,623
274,815 105,000 223,639 78,525
300,290 106,639 248,213 79,344
324,638 108,525 275,491 80,000
350,789 109,918 299,614 82,377
376,039 109,426 324,412 83,525
399,710 110,738 350,564 84,508
426,312 112,541
450,660 112,787
DYNAMIC MODULUS OF ASPHALT CONCRETE Temperature
(C) 4,0
Temperature
(C) 20,0
Temperature
(C) 40,0
Frequency
(Hz)
Dynamic
Modulus
(MPa)
Phase Angle
()
Frequency
(Hz)
Dynamic
Modulus
(MPa)
Phase Angle
()
Frequency
(Hz)
Dynamic
Modulus
(MPa)
Phase Angle
()
2,50E+01 1,87E+04 5,65E+00 2,50E+01 9,92E+03 1,41E+01 2,50E+01
2,89E+03 2,82E+01
1,00E+01 1,74E+04 6,66E+00 1,00E+01 8,52E+03 1,58E+01 1,00E+01
2,27E+03 2,96E+01
5,00E+00 1,64E+04 7,40E+00 5,00E+00 7,43E+03 1,74E+01 5,00E+00
1,83E+03 3,03E+01
1,00E+00 1,38E+04 8,97E+00 1,00E+00 5,33E+03 2,09E+01 1,00E+00
1,10E+03 3,17E+01
5,00E-01 1,28E+04 9,86E+00 5,00E-01 4,53E+03 2,26E+01 5,00E-01
9,03E+02 3,12E+01
1,00E-01 1,05E+04 1,27E+01 1,00E-01 3,08E+03 2,69E+01 1,00E-01
6,20E+02 3,08E+01
-
Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P
CILAMCE 2014
Proceedings of the XXXV Iberian Latin-American Congress on
Computational Methods in Engineering
Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil,
November 23-26, 2014
ACKNOWLEDGEMENTS
The authors acknowledge Eletrobras Furnas for sponsoring the
project on thermo-quimo-
mechanical modeling and optimization of concrete dams at UFC,
contract No. 9000000594.
The third author also acknowledges financial support from
"Programa de Recursos Humanos
da ANP para o Setor Petrleo e Gs PRH31-ANP/MCT".
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