Page 29 - Mechanics of Asphalt Microstructure and Micromechanics

22 Ch a p t e r O n e

Please note that n l e k represents the nine independent bases and can be replaced as
n j e i . By replacing l with j and k with i, one has:
σ ne = T T σ ne

ij j i rj si rs j i
σ = TT σ

rs sr rj ij (1-105)
In matrix format, it can be represented as:
σ
σ = TT T (1-105a)

1.6.3.4 Stress Invariants
Following Section 2.5, the three invariants of the stress tensor are:

I = σ = trσ (1-106)
1 ii
1
tr )
I = (σσ −σσ ) = 1 ⎡( σ 2 − tr(σ 2 ⎤ ) ⎦ (1-107)
⎣
2 ii jj ij ji
2 2
I = ξσσ σ = det σ (1-108)
3 ijk i 1 j 2 3 k
1.6.3.5 Symmetry
Due to the non-existent distributed momentum (the second principle of Cauchy Stress),
the stress tensor thus defined will be symmetric. Section 3.4.5 will present a proof.

1.6.3.6 Principal Stresses
For symmetric real valued tensors, there exist three directions where the stress vectors
are coincident with the direction of the normal of the planes. These three stresses are the
principal stresses. One can use Equations 1-31, 1-32, and 1-34 to find the principal direc-
tions and principal stresses.

1.6.3.7 Maximum and Minimum Stresses
(n)
As the stress vector is t = n ●s, its component along the normal of the surface is:
σ = t n () • n = • σ n = σ n n (1-109)
•
N
n
ij i j
Note that s N is the magnitude of the stress vector projected on the normal direction.
The magnitude of the stress vector projected on the tangent direction is:
σ = t n () t • n () − σ 2 (1-110)
2
T N
The maximum or minimum are applicable to s N or s T . Taking s N as an example, to
find maximum or minimum s N is equivalent to find the maximum or minimum of the
function:
fn () = σ n n
i ij i j (1-111)
The Lagrangian Multiplier method can be used to find the solution. One can consti-
tute an equation g n () = σ ij i j ( nn − 1 with n i n i −1 = 0. With the necessary condition
nn −ζ
)

i
i i
of the extremal conditions, one will have: