Page 31 - Mechanics of Asphalt Microstructure and Micromechanics

24 Ch a p t e r O n e

Considering the area transform identity:
ndS = X JN dS o
i A i , A
df = σ n dS = σ X JN dS o
i ji j ji A j , A
P 1 st = σ X J
Ai ji A j ,
Since
X = F −1
Aj ,
One has
−
1
P 1 st = σ F J (1-119)
Ai ji Aj
The second Piola-Kirchhoff stress tensor refers a measure of the forces in the refer-
ence frame to the area in the reference frame.
df = P 2 nd N dS o
o
B AB A
Considering the unit normal n versus N as vector element in the domain, one would
have:
df = F • df
i iB B
df = σ n dS = σ X JN dS = F • df = F • P 2 nd d NdS o
o
i ji j ji A j , A iB B iB AB A
Therefore:
1 −
1 −
P 2 nd = F σ F J (1-120)
AB iB ij Aj
1.6.4 Fundamental Continuum Mechanics Equations

1.6.4.1 General Concepts
There is a set of equations that a continuum must observe during its deformation pro-
cess. These equations include the conservation equations involving mass, momentum,
angular momentum, energy; the second Law of Theromodyanmics; free energies; the
objectivity assumptions; the strain compatibility conditions; and constitutive laws.
The divergence theorem is widely used in derving the differential format of the
conservation laws. It is represented in the following three formats in terms of a scalar
field, a vector field, and a tensor field.
a ∂ ∂A
A
∫∫∫ V η dV = ∫∫ ηn dS ∫∫∫ V x∂ i dV = ∫∫ a n dS ∫∫∫ V x ∂ ij dV = ∫∫ A n dS (1-121a, b, c)
ii
ij i
'i
i
∂ V i ∂ V i ∂ V
1.6.4.2 Density Definition
Considering a volume element ΔV surrounding point P, the amount of mass contained
in ΔV is Δm, the average density of this volume element is:
Δ m
ρ = (1-122)
ave Δ V
When ΔV approaches infinitesimal, it becomes the density r at point P. In reality,
the infinitesimal may not be meaningful when the volume element is smaller than a few
molecules or atoms.