Page 30 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 23

∂g = σδ n +σδ n − ζδ2 n = 0
∂n ij ik j ij jk i ik i (1-112)
k
Considering the symmetry of the stress tensor and the replacing function of the
Delta operator, one can have:
σδ n = σ n σδ n = σ n
ij ik j kj j ij jk i ik i
Therefore:
(σ − ζδ )n = 0 (1-113)
ik ik i
Comparing it with Equation 1-34, it is the same equation. Therefore, the principal
stresses are actually the maximum and minimum stresses. Following the similar proce-
dure, one can determine the maximum shear stresses.
1.6.3.8 Deviatoric and Hydraulic Stresses
The mean of the sum of the three normal stresses ( I = σ = trσ ) is the hydraulic stress
1 ii
or spherical stress. It can be represented as an isotropic stress tensor:
1
σ M = δ I (1-114)
ij ij
3
The difference between the stress tensor and its mean stress tensor is called the de-
viatoric stress tensor.
S = σ −σ M (1-115)
ij ij ij
1.6.3.9 Octohedral Stress
The plane whose normal makes equal angle to the directions of the principal stresses is
called the octahedral plane. The directional cosines of this plane are equal to 1/ 3 and
therefore σ = 1 σ
N
3 ii 1
2
2
σ = σ = ⎡ ⎣ σ − σ ⎤ + ⎡ ⎣ σ − σ ⎤ + ⎡ ⎣ σ − σ ⎤ ⎦ 2 (1-116)
⎦
⎦
T oct 3 1 2 2 3 3 1
Where s 1 s 2 s 3 are the three principal stresses.
1.6.3.10 Piola-Kirchhoff Stresses
In calculating the engineering stresses, one often uses the force applied on a surface
normalized with the surface area in the reference configuration (the original surface).
The first Piola- Kirchhoff (P-K) stress is such a measure.
Δf df
lim i o = i o = p N (1-117)
i
o
ΔS →0 ΔS dS
It should be noted that the direction N is also a measure in the reference frame. The
first P-K stress is defined in such a way that:
p = P N (1-118)
st
N
1
i Ai A
df = σ n dS
i ji j
df = P N dS o
1
st
i Ai A