Page 26 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 19

or ·
αα = Ln n
/
ij i j
)
Ln n = ( D + W n n = D n n
ij i j ij ij i j ij i j
·
•
/
αα =•nDn (1-93)
or ·
αα = Dn n
/
ij i j
1.6.2.7 Rotation Tensor and Stretch Tensor
The deformation gradient tensor F can be decomposed into a rotational component R
followed by a stretch component U or stretch component tensor V followed by a rota-
tional R. It is named as polar decomposition. Figure 1.4 illustrates this concept. PQ is
deformed into pq but can be considered as first stretched to pq (in parallel to PQ) and
then rotated to pq. R is the orthogonal rotational tensor and U and V are symmetric,
positive definite right and left stretch tensors.
dX is stretched to dx and is then rotated to dx. Then one can have:
•
dx = U dX (1-94)

•
dx = R dx
(1-95)
Therefore dx = R U dX and F = R U. Similarly, one can obtain F = V R.
●
●
●
●
Therefore:
•
•
F = R U = V R (1-96)

•
•
•
Since R is purely rotational, one has dx dx = ( R dx •) ( R dx =) dx R Rdx = T dx • dx
T
and R R = I.
Through this relationship and the dot product and strain tensor definitions:
One can prove that:
● T
U U = C and F F = V V (1-97a, 1-97b)
●
●
Reference Configuration
FIGURE 1.4 Illustration of e 3
polar decomposition.
P Q
q´
p
q
x Current Configuration
0
e 2

e 1