Page 442 - Mechanics of Asphalt Microstructure and Micromechanics

434 C hapter T h ir te en

In the atomistic simulations, the information about the position of every atom is readily
available. The virial strain is relatively straightforward. The following is the general
formulation.
The following tensor is defined for atom l:
N ⎛ ΔΔ x ⎞
kl
kl
1 x
q = ∑ ⎜ i j ⎟ (13-3)
l
ij 2
N k ⎝ r ⎠
=1
0
Where Δx = x − x and Δx = x − x . The quantity N refers to the number of near-
k
k
kl
l
kl
l
i i i j j j
est neighbors considered. The left Cauchy-Green strain tensor is given by:
N ⎛ ΔΔ x ⎞
kl
kl
x
b = N q = 1 ∑ ⎜ i j ⎟ (13-4)
l
l
ij λ ij N k ⎝ r 2 ⎠
=1 0
Where l is a factor depending on the lattice being considered.
This definition provides an expression for a measure of deformation defined using
continuum mechanics and in terms of atomic positions. The Eulerian strain tensor of
ij)
1
atom l is obtained from equation, e = ( δ − b . Unlike the virial stress, the atomic
l
l
ij ij
2
strain is valid instantaneously in space and time.
13.2.5.4 Statistical Definition of Stress for Macro Continuum Model
Uncoupled simple constitutive laws including linear elasticity, viscoelasticity, and placid-
ity can be derived from thermomechanics through free energy formulations and can be
derived from the state variables in the MD simulations. For example, the average stress
of a specific region in the atomistic simulation, s ij , on the j plane and in the i direction is
calculated via virial formula (Allen and Tildesley, 1987; Haile, 1992; Chen, 2006):
(13-5)

Where m i is the mass of the atom i, V i is the volume assigned to atom i, N s is the
number of atoms contained in the region of atomic interaction, r ij is the vector from
atom j to atom i, and F ij is that force acting on atom i due to interaction with atom j. The
first term of the right-hand side of Equation 13-5 represents the kinetic effect associated
with atomic motion, and is related to temperature. The second term represents the ef-
fect of atomic interaction and is related to the interatomic forces and the separation be-
tween the atoms.
13.2.5.5 Free Energy and Elasticity
Elasticity stems from the interactions of atoms, and thus it is intimately linked to elec-
tro-chemistry. Depending on what material is being considered, these chemical interac-
tions are more or less complex. In amorphous bitumen materials, it is vital to consider
various types of chemical interactions, including:
• Ionic interactions (electro-static columbic interactions)
• Covalent bonds (due to overlap of electron orbital)