Page 182 - Mechanics of Asphalt Microstructure and Micromechanics

174 Ch a p t e r S i x

Where g is shear deformation. Thus, neo-Hookean solid shows linear dependence of
shear stresses upon shear deformation and quadratic first difference of normal stresses.
These models are not well utilized in AC. They may be well suited for modeling the
large strain elastic behavior of modified asphalt binder.

6.1.3 Crystal Elasticity
Assume there is a collection of N atoms and the existence of potential energy represent-
th
ing the many-body interactions. The force acting on the i atom due to the other N-1
atoms can be expressed as:
∂
f =− Φ( r ,.., r ) (6-49)
i r ∂ 1 N
i
Where r 1 ,..,r N represent the position vectors of N atoms and ƒ i is the force vector. The
simplest case involves the interaction of two atoms. In this situation the popular Lena-
rd-Jones potential may be used, although it may not be correct for most of the metals.
σ 12 σ 6
Φ () = 4ε ( − ) (6-50)
r
LJ r 12 r 6
Where e,s are material constants. One of the atoms is located at the origin of the
frame. The force acting on the two atoms is the same and equal to:
∂ σ 12 σ 6
r = 24ε
f =− Φ () (2 − ) (6-51)
r ∂ LJ r 13 r 7
The equilibrium position is where the force is zero or the repulsive and attractive
forces are equal to each other, d e = 2 s. In other words, this corresponds to the zero-stress
6
states in the material.
Now considering the force induced due to either pulling apart or contracting together
by a distance dr from the equilibrium position, the force would be equal to df and df.
σ 12 σ 6
ε
(
df =−24 26 − 7 ) dr (6-52)
=
r 14 r 8 rd e
It may be stated that at equilibrium position, for very small changes (distance or
strains), the force is linearly proportional to the distance. This is the basis for the linear
elasticity.
For many-body problems, a simple case is that the overall potential is a sum of all
the two body potentials.
−
∑
N ∑
r
,
Φ ( ,.., r ) = N N 1 Φ ( r r ) (6-53)
LJ 1 i 1 ji ≠ LJ i j
=
6.2 Plasticity (Bower, 2010)
6.2.1 General Concept
From Figure 6.1, one can see the effect of yielding for the 1D case. When the total
strain increases, the material cannot take up additional load or stress (or some more
loads for the hardening materials). One can no longer use the elastic constitutive