Page 15 - Mechanics of Asphalt Microstructure and Micromechanics

8 Ch a p t e r O n e

Obviously, the inner product ends up with a vector. The physical meaning is the
projection of a tensor on the surface to give the surface traction (see the stress-surface
traction relationship).

(ab + a b + a b )c e + (ab + a b + a b )c e (1-15)
11 2 2 3 3 2 2 11 2 2 3 3 3 3
Other vector dyad products may include:
(1-16)

(1-17)

1.5.3.5 Dyad-Dyad Product
The dyad-dyad product of two dyads is defined as:
(1-18)
It is a dyad.
1.5.3.6 Vector-Tensor Products

(1-19)

(1-20)
If A is symmetric, it can be conveniently verified that the above two products are
the same.
This product carries the meaning of the projection of a tensor (stress) to the surface
as surface traction. In that case, A is the stress tensor and v is the direction normal of the
surface.

1.5.3.7 Tensor-Tensor Product
(1-21)
The product is a tensor. It follows the same rule as the dyad-dyad product in
1.5.3.5.

1.5.4 Frame Transformation
A scalar will not change its value when the frame is changed, or it is frame independent.
However, vectors and tensors will follow certain rules when the frame is changed.
1.5.4.1 Vector Transformation
Considering two-frame systems (x 1 , x 2 , x 3 ) and (x 1 , x 2 , x 3 ) (Figure 1.2), their basis vectors
are corresponding (e 1 , e 2 , e 3 ) and (e 1 , e 2 , e 3 ). From algebra geometry, one can represent the
relationship of the two basis systems, or the coordinates of the basis vectors in the orig-
inal frame in the following equation:
−
⎧e ′ ⎫ ⎧ cos( 11),cos( 1' − 2),cos(('13− ) ⎫⎧e ⎫
'
1
⎪ ⎪ ⎪ ⎪⎪ ⎪
1
−
−
⎨
2
2
)
⎨ e ′ ⎬ = cos( '2 − 1 ),cos( ' 2 ),cos( ' 3 ⎬⎨ e ⎬
2
2
⎪
⎪ ⎪ ⎪ 3'− 3'− ⎪⎪ ⎪
3
1
⎩ ⎭ ⎩ cos( '− 1),cos( 2),cos( 3) ⎭ ⎩ e 3 ⎭
′ e
3