Page 19 - Mechanics of Asphalt Microstructure and Micromechanics

12 Ch a p t e r O n e

1.5.7 Divergence Theorem
The Integral Theorems of Gauss and Stokes (or the divergence theorem) is widely used
in continuum mechanics. It presents the relationship between the volume integral and
the surface integral of any quantity. For a tensorial quantity T ij…k and its gradient
∂ T
T = ij k
...
...
ij k q , x ∂
q
The theorem states:
∫ T ij k n dS = ∫ T ij k q , dV (1-39)
q
...
...
S V
For a scalar field l, it ends with:
∫ λndS = ∫ λ dV (1-40)
q
q ,
S V
For a vector field v:
q ∫
∫ v ndS = ∫ divvdV or v n dS = ∫ v dV (1-41)
•
qq ,
q
S V S V
v ∂
∫
∫ n vdS = ∫ curlvdV or ∫ ξ ijk n v e dS = ξ ijk x ∂ j edV (1-42)
×
j k
i
k
S V S V i i
In 2D spaces, it reduces to the area integral and curve (line) integral.
1.6 Fundamentals of Continuum Mechanics

1.6.1 Concept of a Continuum and Representative Element
With unaided eyes, metals, glasses, other solids, and liquids look like a continuum;
there is no space in them. One can imagine that the gases are filled with molecules ev-
erywhere and can also be considered a continuum. The physical concept of continuum
is very complicated. There are always voids and minute cracking in solids, and voids in
liquids. There are also spaces between atoms and molecules. Even within the atom
there are tremendous spaces. However, there might be force fields in these spaces which
bond the molecules and atoms. The concept of a continuum in continuum mechanics is
both concrete and abstract. In the abstract sense, when the effects due to any discontinu-
ity are negligible, the material may behave like a continuum. Mathematically, if the
displacement field of a material can be represented as a continuous function, the mate-
rial can be considered a continuum. While there is no restriction on the homogeneity
assumption of the material, this assumption is usually implied as typically one ad-
dresses the material with the same set of constitutive equations. AC is, by nature, a
heterogeneous material comprised of asphalt binder, aggregates, and voids. Each of the
constituents can be considered as homogeneous. Using a large representative volume
element (RVE), the thus obtained properties of RVE can be used to represent the proper-
ties of the material at any sizes, including the infinitesimal elements.