Page 172 - The Combined Finite-Discrete Element Method

CONSTITUTIVE LAW 155

The matrix of the Green–St. Venant strain tensor is therefore given as follows:
(s 1 − 1)/2 0 0
 2 
1
2
E = (B − I) =  0 (s 2 − 1)/2 0  (4.126)
2 2
0 0 (s 3 − 1)/2
In a similar way, the matrix of the logarithmic strain tensor strain tensor is given by

 
ln(s 1 ) 0 0
E o =  0 ln(s 2 ) 0  (4.127)
0 0 ln(s 3 )

The left stretch tensor can be expressed as a composition of shape changing stretch
followed by volume changing stretch:

   
s 00 s 1 /s 0 0
V = V s V d =  0 s 0   0 s 2 /s 0  (4.128)
0 0 s 0 0 s 3 /s

This represents the stretching of elemental volume, as shown in Figure 4.11, where the ele-
mental volume of edge a (infinitesimally small number) is considered. It can be observed
from the figure that the three edges of the initial volume have stretched by a factor s 1 ,s 2
and s 3 , respectively. Thus, the elemental volume is subject to strain in each direction. The
Green–St. Venant strain tensor quantifies this strain as given by equation (4.126).
The same elemental volume has changed its volume. The volume change can be
expressed as the ratio
s 1 a · s 2 a · s 3 a
s v = = s 1 s 2 s 3 (4.129)
a 3
The matrix of the logarithmic strain tensor is calculated from stretch components
as follows:
   
ln(s 1 /s) 0 0 ln(s) 0 0
E o =  0 ln(s 2 /s) 0  +  0 ln(s) 0  (4.130)
0 0 ln(s 3 /s) 0 0 ln(s)

as 3
a
as 2
a

a

as 1

Figure 4.11 Stretching of elemental volume in three orthogonal directions.