Page 164 - MODELING OF ASPHALT CONCRETE

142 Cha pte r S i x

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Being a complex function E is composed of real and imaginary components,
referred to as the storage and loss moduli, respectively, and mathematically expressed
as follows:
E = E +′ iE′′ (6-12)
*
*
where E′= E cosφ storage modulus
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E″= E sinφ loss modulus
E = E () + E ( ″ ) 2
′
2
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i =−1

Determination of LVE Response Functions
Viscoelastic response functions can be determined either through experimental testing
conducted in the LVE range or through interconversion from other known response
functions. From the theory of viscoelasticity, it can be shown that all LVE response
functions are interrelated; thus, any function can be obtained if another is known.
Both the creep compliance and complex modulus tests are simple mechanical tests
that allow for the accurate characterization of AC in the LVE range. From the creep test
D(t) is determined as a function of time; whereas, from the complex modulus test |E |
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and f are determined as a function of frequency. The simplicity of obtaining D(t) and E *
from mechanical tests is countered with a difficulty in obtaining E(t) from the relaxation
test which is more difficult to conduct and requires a high capacity and robust testing
machine. Therefore, it is often the case where E(t) is obtained through interconversion
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from D(t) or E .
Interconversion can also be necessary where one material function cannot be
determined from a single test type over the entire range of the domain needed. For
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example, E(t) and D(t) cannot be determined at very short times; in this case, E is
determined by conducting a complex modulus test for the corresponding range of
interest in frequency domain and then converted to E(t) and D(t). Prior to elaborating
on the different interconversion techniques, it is necessary to present and discuss the
analytical representation of the response functions since that will impact the choice and
accuracy of the interconversion method used.

Analytical Representation of LVE Response Functions
For accurate material characterization to be achieved, it is essential that representative
analytical expressions of LVE response functions be established regardless of how those
functions are obtained. For example, if an analytical expression is to be established for
E , complex modulus tests are first conducted at several temperatures and frequencies.
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The time-temperature superposition principle is then applied to obtain a single
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mastercurve for |E | and f as a function of reduced frequency at a reference temperature
of choice. This has been covered in more depth in Chap. 4. A mathematical function is
then fit to the mastercurve to arrive to a representative analytical expression of that
response for a broad frequency (time) range.