Modeling of complex modulus of bituminous mixtures measured in tension/compression to estimate secant modulus in indirect tensile test

Abstract

For mechanistic based pavement design, it is necessary to know the stiffness of materials used for road construction. In the European standard EN 12697-26, several experimental tests are proposed to measure the modulus of bituminous mixtures. However, bituminous materials exhibit strong viscoelastic behavior. Hence, the stiffness of this specific material depends not only on the sample geometry but also on the loading law (strain or stress curve versus time). As a consequence, comparison between these different tests, performed in time or frequency domain, is not straight forward. The present paper focuses on measurement of secant modulus using the Indirect tensile (IT) test and comparison with complex modulus. In the IT test, the loading law is not systematically controlled. So, it is important to investigate the influence of the loading waveform on the test result. For this study, the case of a High Modulus Bituminous Mixture for base course (French EME) has been considered. Its viscoelastic behavior has been firstly determined using complex modulus measurements and using Prony series model. An intermediate step, based on an original master curve construction method is used. Doing so, the viscoelastic model is validated on the time domain by simulating direct tensile tests. The case of IT test is then considered. The Prony series model is conformed against IT test results for which the force waveform is known. For this last test, the dependance of the Poisson ratio with temperature and loading time is highlighted. Assuming a purely elastic behavior in isotropic compression, a formula, able to derive the viscoelastic Poisson ratio from the complex modulus is presented. Finally, a theoretical parametric study considering the IT loading waveform is undertaken. It appears that the correction factor given in the standard EN 12697-26 cannot be applied at all temperatures. Actually, this correction factor should be material dependent. This study has been carried out in the framework of collaboration between LCPC and USIRF (Union des Syndicats de l’Industrie Routière française).

Appendix 1: Poisson ratio in the case an elastic response of the material under isotropic loading condition

Hypothesis: The viscoelastic response of the material is assumed to originate only from shear strains and stresses whereas the volumetric response is supposed to be only purely elastic. Such an assumption derives from the fact that under a constant compression state of stress, only shear strain can significantly evolve in a long duration, whereas it is not possible for volumetric negative strain.

• The strain/stress law writes:

  $$ \varepsilon^{*} = {\frac{{\sigma^{*} }}{{2\mu^{*} }}} + {\text{tr}}(\sigma^{*} )\left[ {\frac{1}{9K} - {\frac{1}{{6\mu^{*} }}}} \right]I $$

  (26)

  with μ* being the complex shear modulus (temperature and frequency dependent) and K being the elastic bulk modulus. For f → ∞, the law ε*/σ* must be purely elastic. So, we can defined E _inf and ν_inf such as:

  $$ \left\{ {\begin{array}{*{20}c} {K = {\frac{{E_{\inf } }}{{3 \cdot \left( {1 - 2\upsilon_{\inf } } \right)}}}} \\ {\mu \left( {f \to \infty } \right) = {\frac{{E_{\inf } }}{{2 \cdot (1 - \upsilon_{\inf } )}}}} \\ \end{array} } \right. $$

  (27)

• Let us look at this law in case of a simple cyclic compression test for which:

  $$ \sigma^{*} = \left[ {\begin{array}{*{20}c} {\sigma_{xx} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] $$

  (28)

  then

  $$ \varepsilon_{xx}^{*} = {\frac{{\sigma_{xx}^{*} }}{{2\mu^{*} }}}* + \sigma_{xx}^{*} \left[ {\frac{1}{9K} - {\frac{1}{{6\mu^{*} }}}} \right] = \sigma_{xx}^{*} \left[ {\frac{1}{9K} + {\frac{1}{{3\mu^{*} }}}} \right] $$

  (29)

  so

  $$ E^{*} = {\frac{{\sigma_{xx}^{*} }}{{\varepsilon_{xx}^{*} }}} = {\frac{{9K\mu^{*} }}{{3K + \mu^{*} }}} $$

  (30)

  Beside

  $$ \varepsilon_{yy}^{*} = \sigma_{xx}^{*} \left[ {\frac{1}{9K} - {\frac{1}{{6\mu^{*} }}}} \right] $$

  (31)

  So, the complex Poisson ratio value is given by:

  $$ \upsilon^{*} = - {\frac{{\varepsilon_{yy}^{*} }}{{\varepsilon_{xx}^{*} }}} = \frac{1}{2} \cdot {\frac{{3K - 2\mu^{*} }}{{3K + \mu^{*} }}} $$

  (32)

• Now, let’s express ν* as a function of E*:

  Equation 30 gives:

  $$ \mu^{*} = {\frac{{3KE^{*} }}{{9K - E^{*} }}} $$

  (33)

  Mixing Eqs. 32 and 33, we get

  $$ \upsilon^{*} = \frac{1}{2} \cdot {\frac{{3K - 2{\frac{{3KE^{*} }}{{9K - E^{*} }}}}}{{{\frac{{3KE^{*} }}{{9K - E^{*} }}} + 3K}}} = \frac{1}{2} - {\frac{{E^{*} }}{6K}} $$

  (34)

  Finally, taking into account Eq. 27, the complex Poisson ratio writes:

  $$ \upsilon^{*} = \frac{1}{2} - \left( {\frac{1}{2} - \upsilon_{\inf } } \right){\frac{{E^{*} }}{{E_{\inf } }}} $$

  (35)

  It can be observed that:

  $$ \upsilon^{*} (f \to o) = \frac{1}{2} - \left( {\frac{1}{2} - \upsilon_{\inf } } \right){\frac{{E_{0} }}{{E_{\inf } }}} \approx 0.5 $$

  (36)

  since E ₀ ≪ E _inf in case of bituminous materials.

  And

  $$ \upsilon^{*} (f \to \infty ) = \frac{1}{2} - \left( {\frac{1}{2} - \upsilon_{\inf } } \right){\frac{{E_{\inf } }}{{E_{\inf } }}} = \upsilon_{\inf } $$

  (37)

Glossary

$\delta_{\text{moy}}^{{(T_{j} ,T_{j + 1} )}} (\omega )$

Arithmetic mean of phase angles measured at frequency ω and at temperatures T _j and T _j+1

C ^ref₁ and C ^ref₂

Coefficient of WLF law for a reference temperature of T _ref

|z*|

Complex number modulus z

ν*

Complex Poisson’s ratio

E* = E1 + iE2

Complex stiffness modulus

E _inf

E*(ω → ∞)

F

Force (N)

f

Frequency (Hz)

E₀, E _i , τ _i

Parameters of generalized Maxwell model

δ

Phase angle (° ou rad)

ν

Poisson’s ratio

ω = 1/2πf

Pulsation (rad/s)

E(t)

Relaxation modulus (MPa)

ε

Strain

σ

Stress

T

Temperature (°C)

h

Thickness of disk in indirect tensile test (m)

a _T (T₁, T₂)

Translation coefficient between isotherms T ₁ and T ₂ written as a function of log(ω). The coefficient a _T is defined as follows:

E*(T₁,ω) = E*(T₂,a _T (T₁,T₂)ω)

In particular a _T (T ₂, T ₂) = 1

z

Variation in disk diameter during indirect tensile test (m)

ν_inf

ν*(ω → ∞)