Page 24 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 17

1.6.2.6 Changes of Segment, Area, and Volume
1.6.2.6.1 Line Segment Change

•
dx = F dX (1-81)
(1-82)

•
L F dX =
dx =• • L dx (1-83)
The material derivative of the dot product of dx:
(1-84)

1.6.2.6.2 Area Change
The area change can be evaluated through an area element composed of two segments
(2)
dX and dX . In the reference configuration the area is:
(1)
dS = ξ dX dX ( ) (1-85)
()
o
2
1
A ABC B C
In the current configuration:
dS = ξ dx dx ( ) = ξ x dX x dX (22)
()
1
2
1
()
i ijk j k ijk j B , B k C , C
x dS = ξ x x x dX dX ( ) 2 = ξ F F F dX dX ( )
2
1
()
() 1
,
i A i ijk i A j B k C B C ijk iAAjB kC B C
,
,
,
ξ FFF = ξ det F = ξ J
ijk iA jB kC ABC ABC
x dS = ξ JdX dX ( ) 2
() 1
iA i ABC B C
,
Multiply both sides of the above equation by X A,q , one obtains:
() 1
( ) 2
xX dS = ξ JdX dX X
iA A q , i ABC B C A q ,
,
xX = δ
iA A q , iq
,
dS = X JdS o
q A q , A (1-86)
· ·
–1
Considering the identity detB = tr(B B ) detB and replace B with F, one can obtain:
●
. . .
det F = tr (F F• −1 )det F = JtrL or J = Jdivv (1-87)

This relationship is important for defining stress in the reference configuration:
dS = ( −1 T • dS o
J F )

dS F = JdS o
•
(1-88)