Page 190 - MODELING OF ASPHALT CONCRETE

168 Cha pte r Se v e n

Elastic-Viscoelastic Correspondence Principle
The stress-strain relationships for many viscoelastic materials can be represented by
elastic-like equations through the use of so-called pseudovariables. This simplifying
feature enables a class of extended correspondence principles to be established and
applied to linear as well as some nonlinear analyses of viscoelastic deformation and
fracture behavior (Schapery 1984). Using these correspondence principles, one can
obtain viscoelastic solutions from their elastic counterparts through a simple conversion
procedure. The usual Laplace transform-based correspondence principle is limited to
LVE behavior with time-varying boundary conditions, whereas the correspondence
principles based on pseudovariables are applicable to both linear and nonlinear
behavior of a class of viscoelastic materials with stationary or time-dependent boundary
conditions. Also, the latter does not require a transform inversion step to obtain the
viscoelastic solutions but rather requires a convolution integral which is much easier to
handle than the inversion step.
Consider a stress-strain equation for linear viscoelastic materials,
ξ ε ∂
ij ∫
σ = E ( ξ τ) τ ∂ kl τ d (7-4)
−
ijkl
o
where s , e = stress and strain tensors
ij kl
E (t) = the relaxation modulus matrix
ijkl
ξ = ta = reduced time
T
T = physical time
a = the time-temperature shift factor
T
t = the integration variable
Equation (7-4) can be written as
σ
R
σ = E ε R or ε = ij (7-5)
ij R kl kl E
R
if we define
ξ
1 ε ∂
ε = ∫ E ( ξ τ) kl τ d (7-6)
−
R
kl E ijkl τ ∂
R
o
where E is termed the reference modulus, which is a constant and has the same dimension
R
as the relaxation modulus E (t). The usefulness of Eq. (7-5) is that a correspondence can
ijkl
be found between Eq. (7-5) and the linear elastic stress-strain relationship. That is, the
equations in Eq. (7-5) take the form of elastic stress-strain equations even though they are
R
actually viscoelastic stress-strain equations. The ε is called the pseudostrain. The
kl
pseudostrain accounts for all the hereditary effects of the material through the convolution
integral. The reference modulus E is introduced here because it is a useful parameter in
R
discussing special material behaviors and introducing dimensionless variables. For
example, if we take E (t) = E in Eq. (7-6), we obtain ε = ε , and Eq. (7-5) reduces to the
R
kl
ijkl
kl
linear elastic equation σ = R E ε or ε = σ E . If we take E = 1 in Eq. (7-6),
ij ijkl kl kl ij ijkl R
pseudostrains are simply the linear viscoelastic stress response to a particular strain
input. For the remainder of this chapter, E is set to one.
R
These observations suggest that if the hysteretic behavior of asphalt concrete is due
to linear viscoelasticity only, the presentation of the hysteretic data in terms of stress