CILAMCE2014-0518_15149

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]

Fitting and interconverting prony series of viscoelastic engineering materials using a computer program

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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)

Silva, H. N., Soares, J. B., Souza, F. V., Gomes, F. M. P

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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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( ) (

)

( ) (

)

(

)

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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( )

( )

(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-


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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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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


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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.


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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)


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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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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 <


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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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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.


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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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(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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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


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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.


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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


CILAMCE 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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