Page 21 - Mechanics of Asphalt Microstructure and Micromechanics

14 Ch a p t e r O n e

Considering:
(dx − (dX ) = dx dx − dX dX = (x dX )(x dX ) −δ dX dX = (x x −δ )dX dX
2
2
)
)
i i A A , i A A , i B B AB A B , i A i ,B AB A B
= (C −δ )dX dX (1-52)
AB AB A B
Where C AB = x i,A x i,B or C = F T ● F in matrix or tensor representation.
The Lagrangian Finite Strain Tensor is defined as:
2E = C −δ (1-53)
AB AB AB
or

2E = C −δ (1-54)
and

2
2
)
(dx − (dX ) = (C −δ )dX dX = 2 E dX dX (1-55)
AB AB A B AB A B
For the one-to-one mapping, one can represent X as a function of x. Therefore:
(dx − (dX ) = δ dx dx − (X dx )(X dx = X X ) dx dx = (δ − c dx dx
) (δ −
2
2
)
)
ij i j A ,i i A , j j i ij A i , A j , i j ij ij i j
−1
c = X X A j , or c = ( F ) T • F ( −1 ) (1-56)
ij
A i ,
The Cauchy Deformation Tensor or Eulerian Finite Strain Tensor is defined as:
I c
2e = δ ij − c or 2e = − (1-57)
ij
ij
Considering two segments dX and dX in the reference configuration, dx and
(1)
(2)
(1)
(2)
dx represent the two segments in the current configuration.
•
dx • dx ( ) = F dX () • • ( ) = dX () 1 • F • • (2) )
1
2
T
2
1
()
F dX
F dX
= dX () • • dX ( ) = dX () • I ( + E)2 • dX ( ) = dX () • dX (2) + dX 1 ( ) • 2 • dX 2 ( ) (1-58)
2
1
1
2
2
1
E
C
If there is no strain, one can have:
)
2
(1)
1
2
2
( )
1
()
1
()
2
dx • dx ( ) = dx dx cos =θ dX () • dX ( ) = dX dX ( ) cosθ ' (1-59)
It means the angles between any two segments remain unchanged.
1.6.2.2 Displacement Representation
Obviously, the displacement of point X can be represented as:
uX ) = x X ) − X
(
(
i A i A i
ux () = x − X x () (1-60a, 1-60b)
A i A A i
Therefore the Langrangian Finite Strain Tensor is:
2E = x x −δ = u ( +δ )( u +δ ) −δ
,
,
,
,
,
AB iA i B AB iA iA i B i B AB B
,
or
2E = u + u + u u (1-61)
AB A B B A i A i B
,
,
,
,