Page 162 - MODELING OF ASPHALT CONCRETE

140 Cha pte r S i x

The brackets { } indicate that the response is a function of the input history. The
homogeneity condition, also referred to as proportionality, essentially states that the
output is directionally proportional to the input, for example, if the input is doubled,
the response doubles as well. The superposition condition states that the response to the
sum of two inputs is equivalent to the sum of the responses from the individual inputs.
For linear viscoelastic (LVE) materials, the input-response relationship is expressed
through the hereditary integral which allows the response to any input history to be
calculated as follows:
t dI
τ
R = ∫ R t(, ) dτ dτ (6-3)
H
−∞
where R = unit response function
H
I = input
t = time of interest
t = integration variable

With a known unit response function, the lower limit of the integration can be
reduced to 0 (just before time zero) if the input starts at time t 0 and both the input
−
and response are equal to zero at t < 0. The value of 0 is used instead of 0 to allow for
−
the possibility of a discontinuous change in the input at t 0. For notational simplicity,
0 is used as the lower limit in all successive equations and should be interpreted as
−
0 unless specified otherwise. Equation (6-3) is applicable to an aging system in which
the time zero is the time of production of the material rather than the time of load
application. In a nonaging system, time zero corresponds to the onset of load application,
irrespective of when the material was produced. In this text, asphalt concrete will be
treated as a nonaging system; thus, Eq. (6-3) reduces to
t dI
τ
R = ∫ R t − ) dτ dτ (6-4)
(
H
0
Types of LVE Response Functions
Several viscoelastic response functions can be used to characterize the LVE behavior of
asphalt concrete (AC), the most fundamental ones being the relaxation modulus E(t),
*
creep compliance D(t), and complex modulus E . The need for different types of response
functions is attributed to a number of factors including type of loading application,
conditions under which the material is characterized, and the experimental testing
difficulties that constrain the determination of those functions.
While those response functions, or linear viscoelastic properties, are fundamental
for characterizing the AC in the linear viscoelastic range, they also serve as key
components in constitutive models that describe the nonlinear behavior of AC under
damage. Additionally, when AC specimens are used in experimental testing, response
functions can be used as “viscoelastic fingerprints” to evaluate the specimen-to-specimen
variation and/or determine if the material has been damaged or not.

Creep Compliance and Relaxation Modulus
The creep compliance D(t) is the ratio of strain response to a constant stress input; while
the relaxation modulus E(t) is the ratio of stress response to a constant strain input. If