Page 280 - Mechanics of Asphalt Microstructure and Micromechanics

272 Ch a p t e r E i gh t

Using the Euler method, this integral could be estimated as follows:
Δε = M :[ σ (1 − θ + σ θ] Δt (8-113)
i
)
n n+1
–1
Where M = h and h is the viscosity tensor.
One can expand s n+1 around s n as a Taylor series and truncate the second and above
terms, and then have the following expression:
σ σ + C :(Δ ε − Δ ε ) (8-114)
i
n+1 n
Thus, one has:
Δε = M :[ σ + θC : ( Δε − Δε )] Δt
i
i
n (8-115)
ε
M : σ Δt + θM C : Δ Δt
:
Δε = n
i
:
I + θM C Δt (8-116)
Where I is a rank 4 unit tensor.
For the strain rate equation, the principle of objectivity requires use of the Jaumann
derivative of Cauchy stress (see Lush et al., 1989), however, for small strain problems
with small rotation, the Cauchy stress can be used.
8.6.4 Implementation of Constitutive Model on ABAQUS
In selecting an FE code, robust algorithm, wide validation, and low cost are critical.
ABAQUS (1984) is a good choice in terms of these criteria. Two of the many functions
included with ABAQUS are User’s Material and User’s Element interfaces which are
designed for the implementation of self-developed constitutive models, and interface
models. Once the constitutive models or interface models are implemented, they be-
come an integrated part of ABAQUS and allow for different material combinations.
ABAQUS is a displacement-driving FE code. In solving the global equilibrium
equations it uses an implicit method. Implementation of constitutive models on
ABAQUS through its user-defined material (UMAT) is not a new exercise Lush et al.,
1989), however, due to the implicit method adopted for global computations, the step
used for time integration might be larger than suitable for local integration (constitutive
integration). Subincrementation was widely used in combination with either the ex-
plicit or implicit method. A robust scheme for the local integration is crucial for accu-
racy and efficiency.
One technical tip involving the update of the Jacobian matrix is proper to mention.
The stress Jacobian J ac or the consistent tangent stiffness is defined as:
Δσ
Jac = (8-117)
Δε
A feature concerning the stress Jacobian as emphasized by Hughes (1985) and Lush
et al. (1989) is that it enhances the rate of convergence in the search for the incremental
deformation that leads to the satisfaction of the momentum balance equation, but in the
end has no effect on the accuracy of the solution. The implication of this feature is the