Page 274 - Mechanics of Asphalt Microstructure and Micromechanics

266 Ch a p t e r E i gh t

A unified viscoplasticity constitutive model with combined isotropic and kinematic
hardening was developed to include these properties. The constitutive model is a very
complicated model that had some realistic performance prediction for AC. With this
constitutive model the analysis of the scaling effect of laboratory simulation tests be-
comes feasible and realistic. This section presents the implementation of the SHRP con-
stitutive model on the FE code ABAQUS, the time integration algorithm, the incremen-
tal constitutive equations, and some validations.

8.6.2 The Constitutive Equations
In the SHRP model, dilatancy and hardening were considered to be related to the ag-
gregate skeleton and were modeled by a nonlinear spring of third order Green hyper-
elastcity (Desai and Siriwardane, 1984) with elastic strain energy up to the fourth order.
Temperature and rate dependence were considered to be associated with the binder
and were modeled by a non-linear dashpot. Residual permanent deformation was
modeled by plasticity of Mises type with associated flow, and combined isotropic and
kinematic hardening. A graphical representation of the model is illustrated in Figure
8.5. It is understood that the viscoelasticity component and the elastoplasticity compo-
nent share the same strain, and the stress is the sum of the stresses of the two compo-
nents. Figures 8.5a and 8.5b respectively represent the model in total and incremental
stress-strain format.

The Elastic Component
The elastic component for both viscoelasticity and elasoplasticty components are the
same. The elastic strain energy is a function of the invariants of the elastic strain tensor.
ε
W() = 1 C I + C I + 1 C I + C I I + C I + 1 C I + C I I + C I I + 1 C I (8-69)
2
3
2
e
2
4
I
2 11 22 3 3 1 41 2 5 3 4 6 1 71 2 8 1 3 2 9 2
e
Where e is the elastic strain tensor and I 1 , I 2 , I 3 are the invariants of the elastic strain
e
tensor. W(e ) is the elastic strain energy.
1
ε
I = trace(), I = ( I − ε e : ε e ), I = det() (8-70)
ε
2
e
e
2
1
1
3
2
And
ε =− i (8-71)
ε ε
e

Where e = the inelastic strain tensor and e is the total strain tensor. C i , i = 1,9 are
i
i
material constants. e = creeping strain in the viscoelasticity component and is plastic
strain in the elastoplasticity component.
The stress tensor is:
σ = ∂W
ij ∂ε e (8-72)
ij
The elasticity tensor is:
∂ 2 W
C = (8-73)
e
ijkl ∂ε ∂ε e
ij kl